VandenBroeck and Keller, Equation (6)

Average Error: 16.7 → 0.5

Time: 12.5s

Precision: binary64

Cost: 32968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -2e+18)
   (* PI l)
   (if (<= (* PI l) 200000000.0)
     (- (* PI l) (/ (/ (tan (* PI l)) F) F))
     (* PI l))))

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+18) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 200000000.0) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else {
		tmp = ((double) M_PI) * l;
	}
	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+18) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 200000000.0) {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	} else {
		tmp = Math.PI * l;
	}
	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+18:
		tmp = math.pi * l
	elif (math.pi * l) <= 200000000.0:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	else:
		tmp = math.pi * l
	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+18)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 200000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	else
		tmp = Float64(pi * l);
	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+18)
		tmp = pi * l;
	elseif ((pi * l) <= 200000000.0)
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	else
		tmp = pi * l;
	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[LessEqual[N[(Pi * l), $MachinePrecision], -2e+18], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 200000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e18 or 2e8 < (*.f64 (PI.f64) l)

   1. Initial program 24.0

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

   2. Taylor expanded in l around 0 32.9

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

   3. Simplified 32.9

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]
      Proof
      (*.f64 (/.f64 l F) (/.f64 (PI.f64) F)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l (PI.f64)) (*.f64 F F))): 62 points increase in error, 25 points decrease in error
      (/.f64 (*.f64 l (PI.f64)) (Rewrite<= unpow2_binary64 (pow.f64 F 2))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in F around inf 0.4

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

   if -2e18 < (*.f64 (PI.f64) l) < 2e8

   1. Initial program 9.0

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

   2. Applied egg-rr 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -20:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2
Error	0.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 28000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Error	13.4
Cost	7640

\[\begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-116}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-178}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Error	13.4
Cost	7640

\[\begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -7 \cdot 10^{-8}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-147}:\\ \;\;\;\;\left(-\ell\right) \cdot \frac{\pi}{F \cdot F}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-179}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	13.4
Cost	7640

\[\begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-116}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-179}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6
Error	13.4
Cost	7640

\[\begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-116}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-144}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-177}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\pi}{F} \cdot \left(-\ell\right)}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7
Error	4.9
Cost	7440

\[\begin{array}{l} t_0 := \pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;\ell \leq 25000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8
Error	0.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 8000000000000:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 9
Error	13.5
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2022329 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))