VandenBroeck and Keller, Equation (6)

Average Error: 16.7 → 1.3

Time: 15.5s

Precision: binary64

Cost: 52681

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+45} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell + \frac{1}{F \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right)\right)\right) \cdot \left(-F\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+45) (not (<= (* PI l) 1000000000000.0)))
   (+
    (* PI l)
    (/
     1.0
     (*
      F
      (*
       (log1p (expm1 (fma (* PI -0.3333333333333333) l (/ (/ 1.0 l) PI))))
       (- F)))))
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

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 tmp;
	if (((((double) M_PI) * l) <= -1e+45) || !((((double) M_PI) * l) <= 1000000000000.0)) {
		tmp = (((double) M_PI) * l) + (1.0 / (F * (log1p(expm1(fma((((double) M_PI) * -0.3333333333333333), l, ((1.0 / l) / ((double) M_PI))))) * -F)));
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	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)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+45) || !(Float64(pi * l) <= 1000000000000.0))
		tmp = Float64(Float64(pi * l) + Float64(1.0 / Float64(F * Float64(log1p(expm1(fma(Float64(pi * -0.3333333333333333), l, Float64(Float64(1.0 / l) / pi)))) * Float64(-F)))));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	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_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+45], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1000000000000.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + N[(1.0 / N[(F * N[(N[Log[1 + N[(Exp[N[(N[(Pi * -0.3333333333333333), $MachinePrecision] * l + N[(N[(1.0 / l), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -9.9999999999999993e44 or 1e12 < (*.f64 (PI.f64) l)

   1. Initial program 23.4

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

   2. Applied egg-rr 23.4

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

   3. Taylor expanded in l around 0 18.5

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\color{blue}{\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) \cdot \ell + \frac{1}{\ell \cdot \pi}}} \]

   4. Simplified 18.5

      \[\leadsto \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right)}} \]
      Proof

      [Start] 18.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) \cdot \ell + \frac{1}{\ell \cdot \pi}} \]
      fma-def [=>] 18.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi, \ell, \frac{1}{\ell \cdot \pi}\right)}} \]
      distribute-rgt-out-- [=>] 18.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\mathsf{fma}\left(\color{blue}{\pi \cdot \left(-0.5 - -0.16666666666666666\right)}, \ell, \frac{1}{\ell \cdot \pi}\right)} \]
      metadata-eval [=>] 18.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\mathsf{fma}\left(\pi \cdot \color{blue}{-0.3333333333333333}, \ell, \frac{1}{\ell \cdot \pi}\right)} \]
      associate-/r* [=>] 18.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \frac{1}{\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \color{blue}{\frac{\frac{1}{\ell}}{\pi}}\right)} \]

   5. Applied egg-rr 14.1

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{-1}{\left(\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right) \cdot F\right) \cdot \left(-F\right)}} \]

   6. Applied egg-rr 0.4

      \[\leadsto \pi \cdot \ell - \frac{-1}{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right)\right)\right)} \cdot F\right) \cdot \left(-F\right)} \]

   if -9.9999999999999993e44 < (*.f64 (PI.f64) l) < 1e12

   1. Initial program 10.3

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

   2. Applied egg-rr 2.1

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+45} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell + \frac{1}{F \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right)\right)\right) \cdot \left(-F\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+45} \lor \neg \left(\pi \cdot \ell \leq 1000000000000\right):\\ \;\;\;\;\pi \cdot \ell + \frac{3}{F \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2
Error	8.3
Cost	26697

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\pi \cdot \ell + \frac{3}{F \cdot \left(\pi \cdot \left(\ell \cdot F\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{\frac{F}{\frac{\ell}{F}}}\\ \end{array} \]

Alternative 3
Error	8.2
Cost	26688

\[\pi \cdot \ell + \frac{1}{F \cdot \left(\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right) \cdot \left(-F\right)\right)} \]

Alternative 4
Error	8.2
Cost	26624

\[\pi \cdot \ell + \frac{\frac{-1}{F}}{\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right) \cdot F} \]

Alternative 5
Error	8.2
Cost	26624

\[\pi \cdot \ell + \frac{\frac{-1}{\mathsf{fma}\left(\pi \cdot -0.3333333333333333, \ell, \frac{\frac{1}{\ell}}{\pi}\right) \cdot F}}{F} \]

Alternative 6
Error	21.4
Cost	13376

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

Alternative 7
Error	21.1
Cost	13376

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

Alternative 8
Error	17.2
Cost	13376

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

Alternative 9
Error	17.2
Cost	13376

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

Alternative 10
Error	17.2
Cost	13376

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

Reproduce

herbie shell --seed 2023066 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))