VandenBroeck and Keller, Equation (6)

Percentage Accurate: 75.7% → 95.2%

Time: 25.8s

Alternatives: 5

Speedup: 1.0×

• could not determine a ground truth

  1. F = 5.80289018514445e+21
  2. l = 1941492496190975.5

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: 75.7% 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: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 10^{+16}:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 1e+16)
    (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))
    (expm1 (log1p (* PI l_m))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 1e+16) {
		tmp = (((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F);
	} else {
		tmp = expm1(log1p((((double) M_PI) * l_m)));
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 1e+16) {
		tmp = (Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F);
	} else {
		tmp = Math.expm1(Math.log1p((Math.PI * l_m)));
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 1e+16:
		tmp = (math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F)
	else:
		tmp = math.expm1(math.log1p((math.pi * l_m)))
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 1e+16)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F));
	else
		tmp = expm1(log1p(Float64(pi * l_m)));
	end
	return Float64(l_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 1e+16], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e16

   1. Initial program 79.4%

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

      1. associate-*l/ 80.8%

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

      2. *-un-lft-identity 80.8%

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

      3. associate-/r* 87.7%

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

   3. Applied egg-rr 87.7%

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

   if 1e16 < (*.f64 (PI.f64) l)

   1. Initial program 59.3%

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

      1. fma-neg 59.3%

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

      2. distribute-lft-neg-in 59.3%

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

      3. sqr-neg 59.3%

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

      4. distribute-neg-frac 59.3%

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

      5. metadata-eval 59.3%

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

      6. distribute-lft-neg-out 59.3%

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

      7. neg-mul-1 59.3%

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

      8. associate-/r* 59.3%

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

      9. metadata-eval 59.3%

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

      10. associate-*l/ 59.3%

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

      11. *-lft-identity 59.3%

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

      12. associate-/l/ 59.3%

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

   3. Simplified 59.3%

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

      1. expm1-log1p-u 54.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\right)\right)} \]

      2. associate-/l/ 54.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot \left(-F\right)}}\right)\right)\right) \]

      3. associate-/r* 54.3%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}}\right)\right)\right) \]

      4. add-sqr-sqrt 23.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{-F} \cdot \sqrt{-F}}}\right)\right)\right) \]

      5. sqrt-unprod 54.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{\left(-F\right) \cdot \left(-F\right)}}}\right)\right)\right) \]

      6. sqr-neg 54.2%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{\color{blue}{F \cdot F}}}\right)\right)\right) \]

      7. sqrt-prod 30.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}\right)\right)\right) \]

      8. add-sqr-sqrt 54.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right)\right) \]

      9. associate-/r* 54.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\right)\right)\right) \]

      10. div-inv 54.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right)\right) \]

      11. pow2 54.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right)\right) \]

      12. pow-flip 54.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right)\right) \]

      13. metadata-eval 54.1%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{\color{blue}{-2}}\right)\right)\right) \]

   5. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{-2}\right)\right)\right)} \]

   6. Taylor expanded in F around inf 91.3%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \ell \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def 91.3%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]

   8. Simplified 91.3%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{+16}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 2: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(\pi, l_m, \frac{\frac{-l_m}{\frac{F}{\pi}}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 10000.0)
    (fma PI l_m (/ (/ (- l_m) (/ F PI)) F))
    (expm1 (log1p (* 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) <= 10000.0) {
		tmp = fma(((double) M_PI), l_m, ((-l_m / (F / ((double) M_PI))) / F));
	} else {
		tmp = expm1(log1p((((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) <= 10000.0)
		tmp = fma(pi, l_m, Float64(Float64(Float64(-l_m) / Float64(F / pi)) / F));
	else
		tmp = expm1(log1p(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], 10000.0], N[(Pi * l$95$m + N[(N[((-l$95$m) / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e4

   1. Initial program 79.3%

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

      1. fma-neg 79.3%

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

      2. distribute-lft-neg-in 79.3%

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

      3. sqr-neg 79.3%

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

      4. distribute-neg-frac 79.3%

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

      5. metadata-eval 79.3%

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

      6. distribute-lft-neg-out 79.3%

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

      7. neg-mul-1 79.3%

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

      8. associate-/r* 79.3%

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

      9. metadata-eval 79.3%

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

      10. associate-*l/ 80.7%

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

      11. *-lft-identity 80.7%

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

      12. associate-/l/ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in l around 0 80.2%

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

      1. mul-1-neg 80.2%

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

      2. associate-/l* 80.2%

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

   6. Simplified 80.2%

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

   if 1e4 < (*.f64 (PI.f64) l)

   1. Initial program 60.3%

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

      1. fma-neg 60.3%

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

      2. distribute-lft-neg-in 60.3%

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

      3. sqr-neg 60.3%

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

      4. distribute-neg-frac 60.3%

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

      5. metadata-eval 60.3%

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

      6. distribute-lft-neg-out 60.3%

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

      7. neg-mul-1 60.3%

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

      8. associate-/r* 60.3%

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

      9. metadata-eval 60.3%

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

      10. associate-*l/ 60.3%

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

      11. *-lft-identity 60.3%

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

      12. associate-/l/ 60.3%

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

   3. Simplified 60.3%

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

      1. expm1-log1p-u 55.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\right)\right)} \]

      2. associate-/l/ 55.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot \left(-F\right)}}\right)\right)\right) \]

      3. associate-/r* 55.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}}\right)\right)\right) \]

      4. add-sqr-sqrt 25.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{-F} \cdot \sqrt{-F}}}\right)\right)\right) \]

      5. sqrt-unprod 55.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{\left(-F\right) \cdot \left(-F\right)}}}\right)\right)\right) \]

      6. sqr-neg 55.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{\color{blue}{F \cdot F}}}\right)\right)\right) \]

      7. sqrt-prod 29.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}\right)\right)\right) \]

      8. add-sqr-sqrt 53.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right)\right) \]

      9. associate-/r* 53.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\right)\right)\right) \]

      10. div-inv 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right)\right) \]

      11. pow2 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right)\right) \]

      12. pow-flip 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right)\right) \]

      13. metadata-eval 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{\color{blue}{-2}}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{-2}\right)\right)\right)} \]

   6. Taylor expanded in F around inf 89.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \ell \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def 89.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]

   8. Simplified 89.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10000:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{-\ell}{\frac{F}{\pi}}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;l_m \leq 3400:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{l_m}{F}}{\frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot l_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= l_m 3400.0)
    (- (* PI l_m) (/ (/ l_m F) (/ F PI)))
    (expm1 (log1p (* 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 <= 3400.0) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) / (F / ((double) M_PI)));
	} else {
		tmp = expm1(log1p((((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 <= 3400.0) {
		tmp = (Math.PI * l_m) - ((l_m / F) / (F / Math.PI));
	} else {
		tmp = Math.expm1(Math.log1p((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 <= 3400.0:
		tmp = (math.pi * l_m) - ((l_m / F) / (F / math.pi))
	else:
		tmp = math.expm1(math.log1p((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 <= 3400.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) / Float64(F / pi)));
	else
		tmp = expm1(log1p(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[l$95$m, 3400.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[N[Log[1 + N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3400

   1. Initial program 79.3%

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

      1. sqr-neg 79.3%

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

      2. associate-*l/ 80.7%

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

      3. sqr-neg 80.7%

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

      4. *-lft-identity 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in l around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

      2. clear-num 80.1%

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

      3. un-div-inv 80.2%

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

   8. Applied egg-rr 80.2%

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

   if 3400 < l

   1. Initial program 60.3%

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

      1. fma-neg 60.3%

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

      2. distribute-lft-neg-in 60.3%

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

      3. sqr-neg 60.3%

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

      4. distribute-neg-frac 60.3%

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

      5. metadata-eval 60.3%

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

      6. distribute-lft-neg-out 60.3%

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

      7. neg-mul-1 60.3%

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

      8. associate-/r* 60.3%

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

      9. metadata-eval 60.3%

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

      10. associate-*l/ 60.3%

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

      11. *-lft-identity 60.3%

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

      12. associate-/l/ 60.3%

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

   3. Simplified 60.3%

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

      1. expm1-log1p-u 55.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\right)\right)} \]

      2. associate-/l/ 55.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot \left(-F\right)}}\right)\right)\right) \]

      3. associate-/r* 55.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{-F}}\right)\right)\right) \]

      4. add-sqr-sqrt 25.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{-F} \cdot \sqrt{-F}}}\right)\right)\right) \]

      5. sqrt-unprod 55.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{\left(-F\right) \cdot \left(-F\right)}}}\right)\right)\right) \]

      6. sqr-neg 55.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{\color{blue}{F \cdot F}}}\right)\right)\right) \]

      7. sqrt-prod 29.5%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}\right)\right)\right) \]

      8. add-sqr-sqrt 53.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\color{blue}{F}}\right)\right)\right) \]

      9. associate-/r* 53.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\right)\right)\right) \]

      10. div-inv 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right)\right) \]

      11. pow2 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right)\right) \]

      12. pow-flip 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right)\right) \]

      13. metadata-eval 53.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{\color{blue}{-2}}\right)\right)\right) \]

   5. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\pi, \ell, \tan \left(\pi \cdot \ell\right) \cdot {F}^{-2}\right)\right)\right)} \]

   6. Taylor expanded in F around inf 89.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \ell \cdot \pi\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def 89.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]

   8. Simplified 89.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\ell \cdot \pi\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3400:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{l_m}{F} \cdot \frac{\pi}{F}\right) \end{array} \]

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) (* (/ l_m F) (/ PI F)))))

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) - ((l_m / F) * (((double) M_PI) / F)));
}

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) - ((l_m / F) * (Math.PI / F)));
}

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) - ((l_m / F) * (math.pi / F)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F))))
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) - ((l_m / F) * (pi / F)));
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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. sqr-neg 75.1%

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

   2. associate-*l/ 76.2%

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

   3. sqr-neg 76.2%

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

   4. *-lft-identity 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. times-frac 74.4%

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

6. Applied egg-rr 74.4%

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

7. Final simplification 74.4%

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

Alternative 5: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\frac{l_m}{F}}{\frac{F}{\pi}}\right) \end{array} \]

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) (/ (/ l_m F) (/ F PI)))))

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) - ((l_m / F) / (F / ((double) M_PI))));
}

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) - ((l_m / F) / (F / Math.PI)));
}

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) - ((l_m / F) / (F / math.pi)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) / Float64(F / pi))))
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) - ((l_m / F) / (F / pi)));
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[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] / N[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

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

   1. sqr-neg 75.1%

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

   2. associate-*l/ 76.2%

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

   3. sqr-neg 76.2%

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

   4. *-lft-identity 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in l around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. times-frac 74.4%

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

6. Applied egg-rr 74.4%

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

   1. *-commutative 74.4%

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

   2. clear-num 74.4%

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

   3. un-div-inv 74.4%

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

8. Applied egg-rr 74.4%

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

9. Final simplification 74.4%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))