VandenBroeck and Keller, Equation (6)

Average Error: 17.1 → 1.2

Time: 10.6s

Precision: binary64

Cost: 26568

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-21}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -1e+18)
   (* PI l)
   (if (<= (* PI l) 1e-21) (- (* PI l) (/ (/ (* 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) <= -1e+18) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 1e-21) {
		tmp = (((double) M_PI) * l) - (((((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) <= -1e+18) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 1e-21) {
		tmp = (Math.PI * l) - (((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) <= -1e+18:
		tmp = math.pi * l
	elif (math.pi * l) <= 1e-21:
		tmp = (math.pi * l) - (((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) <= -1e+18)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 1e-21)
		tmp = Float64(Float64(pi * l) - Float64(Float64(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) <= -1e+18)
		tmp = pi * l;
	elseif ((pi * l) <= 1e-21)
		tmp = (pi * l) - (((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], -1e+18], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 1e-21], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e18 or 9.99999999999999908e-22 < (*.f64 (PI.f64) l)

   1. Initial program 23.3

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

   2. Simplified 23.3

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      Proof
      (-.f64 (*.f64 (PI.f64) l) (/.f64 (tan.f64 (*.f64 (PI.f64) l)) (*.f64 F F))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (PI.f64) l) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (tan.f64 (*.f64 (PI.f64) l)))) (*.f64 F F))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (PI.f64) l) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))): 5 points increase in error, 6 points decrease in error

   3. Taylor expanded in l around inf 1.4

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

   if -1e18 < (*.f64 (PI.f64) l) < 9.99999999999999908e-22

   1. Initial program 10.0

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

   2. Applied egg-rr 0.6

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

   3. Taylor expanded in l around 0 1.0

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-21}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8055912406.41709:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{-21}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2
Error	14.1
Cost	7376

\[\begin{array}{l} t_0 := \pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{-127}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-225}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Error	14.1
Cost	7376

\[\begin{array}{l} \mathbf{if}\;F \leq -1.32 \cdot 10^{-127}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-191}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-225}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-134}:\\ \;\;\;\;\pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Error	1.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;\ell \leq -47000:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{-21}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	13.5
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2022325 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))