VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 98.8%

Time: 15.2s

Alternatives: 10

Speedup: 1.5×

• could not determine a ground truth

  1. F = 5.337024520052709e+246
  2. l = 1976673924960277.5

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: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{\frac{\tan \left(\pi \cdot l_m\right)}{-F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1000000.0)
    (fma 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) <= 1000000.0) {
		tmp = fma(((double) M_PI), l_m, ((tan((((double) M_PI) * l_m)) / -F) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1000000.0)
		tmp = fma(pi, l_m, Float64(Float64(tan(Float64(pi * l_m)) / Float64(-F)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1000000.0], N[(Pi * l$95$m + 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) < 1e6

   1. Initial program 74.7%

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

      1. fma-neg 74.7%

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

      2. distribute-lft-neg-in 74.7%

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

      3. sqr-neg 74.7%

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

      4. distribute-neg-frac 74.7%

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

      5. metadata-eval 74.7%

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

      6. distribute-lft-neg-out 74.7%

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

      7. neg-mul-1 74.7%

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

      8. associate-/r* 74.7%

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

      9. metadata-eval 74.7%

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

      10. associate-*l/ 76.0%

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

      11. *-lft-identity 76.0%

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

      12. associate-/l/ 84.8%

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

   3. Simplified 84.8%

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

   if 1e6 < (*.f64 (PI.f64) l)

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 47.3%

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

   6. Taylor expanded in F around inf 99.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 1000000:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1000000.0)
    (- (* 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) <= 1000000.0) {
		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) <= 1000000.0) {
		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) <= 1000000.0:
		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) <= 1000000.0)
		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) <= 1000000.0)
		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], 1000000.0], 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) < 1e6

   1. Initial program 74.7%

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

      1. associate-*l/ 76.0%

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

      2. *-un-lft-identity 76.0%

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

      3. associate-/r* 84.8%

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

   4. Applied egg-rr 84.8%

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

   if 1e6 < (*.f64 (PI.f64) l)

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 47.3%

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

   6. Taylor expanded in F around inf 99.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 1000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{\frac{-l_m}{\frac{F}{\pi}}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 400000.0)
    (fma PI l_m (/ (/ (- l_m) (/ F PI)) F))
    (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 400000.0) {
		tmp = fma(((double) M_PI), l_m, ((-l_m / (F / ((double) M_PI))) / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 400000.0)
		tmp = fma(pi, l_m, Float64(Float64(Float64(-l_m) / Float64(F / pi)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 400000.0], N[(Pi * l$95$m + N[(N[((-l$95$m) / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e5

   1. Initial program 74.7%

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

      1. fma-neg 74.7%

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

      2. distribute-lft-neg-in 74.7%

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

      3. sqr-neg 74.7%

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

      4. distribute-neg-frac 74.7%

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

      5. metadata-eval 74.7%

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

      6. distribute-lft-neg-out 74.7%

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

      7. neg-mul-1 74.7%

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

      8. associate-/r* 74.7%

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

      9. metadata-eval 74.7%

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

      10. associate-*l/ 76.1%

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

      11. *-lft-identity 76.1%

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

      12. associate-/l/ 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in l around 0 83.8%

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

      1. mul-1-neg 83.8%

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

      2. associate-/l* 83.8%

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

   7. Simplified 83.8%

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

   if 4e5 < (*.f64 (PI.f64) l)

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 46.7%

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

   6. Taylor expanded in F around inf 98.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 400000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{-\ell}{\frac{F}{\pi}}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 400000:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{l_m}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 400000.0)
    (- (* PI l_m) (/ (/ l_m (/ F PI)) F))
    (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 400000.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m / (F / ((double) M_PI))) / F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 400000.0) {
		tmp = (Math.PI * l_m) - ((l_m / (F / Math.PI)) / F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 400000.0:
		tmp = (math.pi * l_m) - ((l_m / (F / math.pi)) / F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 400000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / Float64(F / pi)) / F));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 400000.0)
		tmp = (pi * l_m) - ((l_m / (F / pi)) / F);
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 400000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e5

   1. Initial program 74.7%

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

      1. associate-*l/ 76.1%

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

      2. *-un-lft-identity 76.1%

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

      3. associate-/r* 84.9%

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

   4. Applied egg-rr 84.9%

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

   5. Taylor expanded in l around 0 83.8%

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

      1. associate-/l* 83.8%

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

   7. Simplified 83.8%

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

   if 4e5 < (*.f64 (PI.f64) l)

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 46.7%

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

   6. Taylor expanded in F around inf 98.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 400000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 90000:\\ \;\;\;\;\pi \cdot l_m - \frac{\pi}{F} \cdot \frac{l_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 90000.0) (- (* PI l_m) (* (/ PI F) (/ l_m F))) (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 90000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / F));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 90000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / F));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 90000.0:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / F))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 90000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / F)));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 90000.0)
		tmp = (pi * l_m) - ((pi / F) * (l_m / F));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 90000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 9e4

   1. Initial program 74.7%

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

      1. sqr-neg 74.7%

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

      2. associate-*l/ 76.1%

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

      3. sqr-neg 76.1%

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

      4. *-lft-identity 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in l around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. times-frac 83.8%

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

   7. Applied egg-rr 83.8%

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

   if 9e4 < l

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 46.7%

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

   6. Taylor expanded in F around inf 98.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 90000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2.65 \cdot 10^{-188} \lor \neg \left(l_m \leq 9 \cdot 10^{-76}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;l_m \cdot \frac{-\pi}{{F}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= l_m 2.65e-188) (not (<= l_m 9e-76)))
    (* PI l_m)
    (* l_m (/ (- PI) (pow F 2.0))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 2.65e-188) || !(l_m <= 9e-76)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = l_m * (-((double) M_PI) / pow(F, 2.0));
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 2.65e-188) || !(l_m <= 9e-76)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = l_m * (-Math.PI / Math.pow(F, 2.0));
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m <= 2.65e-188) or not (l_m <= 9e-76):
		tmp = math.pi * l_m
	else:
		tmp = l_m * (-math.pi / math.pow(F, 2.0))
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((l_m <= 2.65e-188) || !(l_m <= 9e-76))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(l_m * Float64(Float64(-pi) / (F ^ 2.0)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m <= 2.65e-188) || ~((l_m <= 9e-76)))
		tmp = pi * l_m;
	else
		tmp = l_m * (-pi / (F ^ 2.0));
	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[Or[LessEqual[l$95$m, 2.65e-188], N[Not[LessEqual[l$95$m, 9e-76]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(l$95$m * N[((-Pi) / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.65000000000000007e-188 or 9.0000000000000001e-76 < l

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. associate-*l/ 70.8%

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

      3. sqr-neg 70.8%

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

      4. *-lft-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in l around 0 63.9%

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

   6. Taylor expanded in F around inf 76.5%

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

   if 2.65000000000000007e-188 < l < 9.0000000000000001e-76

   1. Initial program 85.1%

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

      1. sqr-neg 85.1%

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

      2. associate-*l/ 91.0%

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

      3. sqr-neg 91.0%

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

      4. *-lft-identity 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in l around 0 91.0%

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

   6. Taylor expanded in F around 0 60.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 60.8%

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

      2. associate-*r/ 55.0%

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

   8. Simplified 55.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.65 \cdot 10^{-188} \lor \neg \left(\ell \leq 9 \cdot 10^{-76}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{{F}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 1.3 \cdot 10^{-188} \lor \neg \left(l_m \leq 4.8 \cdot 10^{-74}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l_m}{{F}^{2}} \cdot \left(-\pi\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= l_m 1.3e-188) (not (<= l_m 4.8e-74)))
    (* PI l_m)
    (* (/ l_m (pow F 2.0)) (- PI)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 1.3e-188) || !(l_m <= 4.8e-74)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = (l_m / pow(F, 2.0)) * -((double) M_PI);
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 1.3e-188) || !(l_m <= 4.8e-74)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = (l_m / Math.pow(F, 2.0)) * -Math.PI;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m <= 1.3e-188) or not (l_m <= 4.8e-74):
		tmp = math.pi * l_m
	else:
		tmp = (l_m / math.pow(F, 2.0)) * -math.pi
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((l_m <= 1.3e-188) || !(l_m <= 4.8e-74))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(l_m / (F ^ 2.0)) * Float64(-pi));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m <= 1.3e-188) || ~((l_m <= 4.8e-74)))
		tmp = pi * l_m;
	else
		tmp = (l_m / (F ^ 2.0)) * -pi;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[Or[LessEqual[l$95$m, 1.3e-188], N[Not[LessEqual[l$95$m, 4.8e-74]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(l$95$m / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] * (-Pi)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e-188 or 4.7999999999999998e-74 < l

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. associate-*l/ 70.8%

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

      3. sqr-neg 70.8%

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

      4. *-lft-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in l around 0 63.9%

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

   6. Taylor expanded in F around inf 76.5%

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

   if 1.3e-188 < l < 4.7999999999999998e-74

   1. Initial program 85.1%

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

      1. fma-neg 85.1%

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

      2. distribute-lft-neg-in 85.1%

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

      3. sqr-neg 85.1%

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

      4. distribute-neg-frac 85.1%

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

      5. metadata-eval 85.1%

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

      6. distribute-lft-neg-out 85.1%

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

      7. neg-mul-1 85.1%

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

      8. associate-/r* 85.1%

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

      9. metadata-eval 85.1%

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

      10. associate-*l/ 91.0%

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

      11. *-lft-identity 91.0%

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

      12. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in l around 0 85.1%

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

      1. mul-1-neg 85.1%

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

      2. unsub-neg 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in F around 0 55.0%

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

      1. *-rgt-identity 55.0%

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

      2. unpow2 55.0%

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

      3. times-frac 55.0%

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

      4. neg-mul-1 55.0%

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

      5. *-rgt-identity 55.0%

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

      6. associate-*r/ 55.0%

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

      7. associate-*r* 55.0%

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

      8. unpow-1 55.0%

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

      9. unpow-1 55.0%

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

      10. pow-sqr 55.2%

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

      11. metadata-eval 55.2%

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

      12. distribute-rgt-neg-in 55.2%

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

   10. Simplified 55.2%

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

   11. Taylor expanded in l around 0 60.8%

       \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
   12. Step-by-step derivation

       1. mul-1-neg 60.8%

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

       2. associate-*l/ 60.8%

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

   13. Simplified 60.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-188} \lor \neg \left(\ell \leq 4.8 \cdot 10^{-74}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{F}^{2}} \cdot \left(-\pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 2 \cdot 10^{-188} \lor \neg \left(l_m \leq 7.2 \cdot 10^{-74}\right):\\ \;\;\;\;\pi \cdot l_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-l_m\right)}{{F}^{2}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (or (<= l_m 2e-188) (not (<= l_m 7.2e-74)))
    (* PI l_m)
    (/ (* PI (- l_m)) (pow F 2.0)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 2e-188) || !(l_m <= 7.2e-74)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = (((double) M_PI) * -l_m) / pow(F, 2.0);
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((l_m <= 2e-188) || !(l_m <= 7.2e-74)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = (Math.PI * -l_m) / Math.pow(F, 2.0);
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (l_m <= 2e-188) or not (l_m <= 7.2e-74):
		tmp = math.pi * l_m
	else:
		tmp = (math.pi * -l_m) / math.pow(F, 2.0)
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if ((l_m <= 2e-188) || !(l_m <= 7.2e-74))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(pi * Float64(-l_m)) / (F ^ 2.0));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((l_m <= 2e-188) || ~((l_m <= 7.2e-74)))
		tmp = pi * l_m;
	else
		tmp = (pi * -l_m) / (F ^ 2.0);
	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[Or[LessEqual[l$95$m, 2e-188], N[Not[LessEqual[l$95$m, 7.2e-74]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(Pi * (-l$95$m)), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.9999999999999999e-188 or 7.2000000000000005e-74 < l

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. associate-*l/ 70.8%

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

      3. sqr-neg 70.8%

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

      4. *-lft-identity 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in l around 0 63.9%

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

   6. Taylor expanded in F around inf 76.5%

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

   if 1.9999999999999999e-188 < l < 7.2000000000000005e-74

   1. Initial program 85.1%

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

      1. sqr-neg 85.1%

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

      2. associate-*l/ 91.0%

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

      3. sqr-neg 91.0%

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

      4. *-lft-identity 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in l around 0 91.0%

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

   6. Taylor expanded in F around 0 60.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-188} \lor \neg \left(\ell \leq 7.2 \cdot 10^{-74}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{{F}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 90000:\\ \;\;\;\;l_m \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (if (<= l_m 90000.0) (* l_m (* PI (- 1.0 (pow F -2.0)))) (* PI l_m))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 90000.0) {
		tmp = l_m * (((double) M_PI) * (1.0 - pow(F, -2.0)));
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if (l_m <= 90000.0) {
		tmp = l_m * (Math.PI * (1.0 - Math.pow(F, -2.0)));
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if l_m <= 90000.0:
		tmp = l_m * (math.pi * (1.0 - math.pow(F, -2.0)))
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (l_m <= 90000.0)
		tmp = Float64(l_m * Float64(pi * Float64(1.0 - (F ^ -2.0))));
	else
		tmp = Float64(pi * l_m);
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if (l_m <= 90000.0)
		tmp = l_m * (pi * (1.0 - (F ^ -2.0)));
	else
		tmp = pi * l_m;
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[l$95$m, 90000.0], N[(l$95$m * N[(Pi * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 9e4

   1. Initial program 74.7%

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

      1. fma-neg 74.7%

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

      2. distribute-lft-neg-in 74.7%

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

      3. sqr-neg 74.7%

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

      4. distribute-neg-frac 74.7%

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

      5. metadata-eval 74.7%

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

      6. distribute-lft-neg-out 74.7%

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

      7. neg-mul-1 74.7%

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

      8. associate-/r* 74.7%

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

      9. metadata-eval 74.7%

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

      10. associate-*l/ 76.1%

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

      11. *-lft-identity 76.1%

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

      12. associate-/l/ 84.9%

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

   3. Simplified 84.9%

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

   5. Taylor expanded in l around 0 73.6%

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

      1. mul-1-neg 73.6%

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

      2. unsub-neg 73.6%

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

   7. Simplified 73.6%

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

      1. div-inv 73.5%

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

      2. pow-flip 73.6%

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

      3. metadata-eval 73.6%

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

      4. cancel-sign-sub-inv 73.6%

         \[\leadsto \ell \cdot \color{blue}{\left(\pi + \left(-\pi\right) \cdot {F}^{-2}\right)} \]

      5. *-un-lft-identity 73.6%

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

      6. *-commutative 73.6%

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

      7. distribute-lft-neg-in 73.6%

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

      8. distribute-rgt-neg-in 73.6%

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

      9. distribute-lft-in 73.6%

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

      10. sub-neg 73.6%

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

      11. *-commutative 73.6%

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

   9. Applied egg-rr 73.6%

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

   if 9e4 < l

   1. Initial program 65.1%

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

      1. sqr-neg 65.1%

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

      2. associate-*l/ 65.1%

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

      3. sqr-neg 65.1%

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

      4. *-lft-identity 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in l around 0 46.7%

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

   6. Taylor expanded in F around inf 98.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 90000:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 - {F}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 3.0× 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 1 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 71.9%

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

   1. sqr-neg 71.9%

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

   2. associate-*l/ 72.8%

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

   3. sqr-neg 72.8%

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

   4. *-lft-identity 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in l around 0 66.6%

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

6. Taylor expanded in F around inf 72.0%

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

7. Final simplification 72.0%

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

Reproduce

herbie shell --seed 2024010 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))