VandenBroeck and Keller, Equation (6)

Average Error: 16.6 → 0.5

Time: 14.5s

Precision: binary64

Cost: 32969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+17) (not (<= (* PI l) 500000.0)))
   (* PI l)
   (- (* 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) <= -2e+17) || !((((double) M_PI) * l) <= 500000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

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

↓

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+17) || !((Math.PI * l) <= 500000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

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

↓

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+17) or not ((math.pi * l) <= 500000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.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) <= -2e+17) || !(Float64(pi * l) <= 500000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+17) || ~(((pi * l) <= 500000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = 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], -2e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 500000.0]], $MachinePrecision]], N[(Pi * l), $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) < -2e17 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

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

      [Start] 23.7	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 23.7	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 23.7	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around inf 0.5

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

   if -2e17 < (*.f64 (PI.f64) l) < 5e5

   1. Initial program 9.1

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

   2. Applied egg-rr 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000000 \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000000 \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	20233

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -40000000 \lor \neg \left(\pi \cdot \ell \leq 500000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell + \frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \end{array} \]

Alternative 4
Error	13.1
Cost	7640

\[\begin{array}{l} t_0 := 1 + \left(\pi \cdot \ell + -1\right)\\ t_1 := \frac{-\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{if}\;F \leq -0.000215:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	13.1
Cost	7376

\[\begin{array}{l} t_0 := \ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{if}\;F \leq -0.000245:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6
Error	4.7
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -122 \lor \neg \left(\ell \leq 780000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \]

Alternative 7
Error	13.0
Cost	6528

\[\pi \cdot \ell \]

Alternative 8
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023237 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))