VandenBroeck and Keller, Equation (6)

Average Error: 16.9 → 0.7

Time: 7.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \ell\right)\\ t_1 := \pi \cdot \ell - {\left(\frac{F}{t_0} \cdot \log \left({\left(e^{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}\right)}^{F}\right)\right)}^{-1}\\ \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{t_0}{F \cdot \cos \left(\pi \cdot \ell\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (sin (* PI l)))
        (t_1
         (-
          (* PI l)
          (pow
           (* (/ F t_0) (log (pow (exp (fma -0.5 (pow (* PI l) 2.0) 1.0)) F)))
           -1.0))))
   (if (<= (* PI l) -2e+21)
     t_1
     (if (<= (* PI l) 500000.0)
       (- (* PI l) (/ (/ t_0 (* F (cos (* PI l)))) F))
       t_1))))

C

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

↓

double code(double F, double l) {
	double t_0 = sin((((double) M_PI) * l));
	double t_1 = (((double) M_PI) * l) - pow(((F / t_0) * log(pow(exp(fma(-0.5, pow((((double) M_PI) * l), 2.0), 1.0)), F))), -1.0);
	double tmp;
	if ((((double) M_PI) * l) <= -2e+21) {
		tmp = t_1;
	} else if ((((double) M_PI) * l) <= 500000.0) {
		tmp = (((double) M_PI) * l) - ((t_0 / (F * cos((((double) M_PI) * l)))) / F);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

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

↓

function code(F, l)
	t_0 = sin(Float64(pi * l))
	t_1 = Float64(Float64(pi * l) - (Float64(Float64(F / t_0) * log((exp(fma(-0.5, (Float64(pi * l) ^ 2.0), 1.0)) ^ F))) ^ -1.0))
	tmp = 0.0
	if (Float64(pi * l) <= -2e+21)
		tmp = t_1;
	elseif (Float64(pi * l) <= 500000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(t_0 / Float64(F * cos(Float64(pi * l)))) / F));
	else
		tmp = t_1;
	end
	return tmp
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]

↓

code[F_, l_] := Block[{t$95$0 = N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * l), $MachinePrecision] - N[Power[N[(N[(F / t$95$0), $MachinePrecision] * N[Log[N[Power[N[Exp[N[(-0.5 * N[Power[N[(Pi * l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], F], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(Pi * l), $MachinePrecision], -2e+21], t$95$1, If[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(t$95$0 / N[(F * N[Cos[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e21 or 5e5 < (*.f64 (PI.f64) l)

   1. Initial program 23.7

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

   2. Simplified 23.7

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

   3. Taylor expanded in l around inf 23.7

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

   4. Simplified 23.7

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

   5. Taylor expanded in l around 0 9.1

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\color{blue}{\left(-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right) + 1\right)} \cdot F}}{F} \]

   6. Applied egg-rr 9.1

      \[\leadsto \pi \cdot \ell - \color{blue}{{\left(\frac{F}{\sin \left(\ell \cdot \pi\right)} \cdot \left(\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right) \cdot F\right)\right)}^{-1}} \]

   7. Applied egg-rr 0.4

      \[\leadsto \pi \cdot \ell - {\left(\frac{F}{\sin \left(\ell \cdot \pi\right)} \cdot \color{blue}{\log \left({\left(e^{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}\right)}^{F}\right)}\right)}^{-1} \]

   if -2e21 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 9.5

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

   2. Simplified 9.1

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

   3. Taylor expanded in l around inf 9.1

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

   4. Simplified 1.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right) \cdot F}}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\pi \cdot \ell - {\left(\frac{F}{\sin \left(\pi \cdot \ell\right)} \cdot \log \left({\left(e^{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}\right)}^{F}\right)\right)}^{-1}\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - {\left(\frac{F}{\sin \left(\pi \cdot \ell\right)} \cdot \log \left({\left(e^{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}\right)}^{F}\right)\right)}^{-1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))