VandenBroeck and Keller, Equation (6)

Average Error: 25.17% → 1.06%

Time: 13.5s

Precision: binary64

Cost: 33097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 0.0002\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

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+17) (not (<= (* PI l) 0.0002)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (tan (* 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+17) || !((((double) M_PI) * l) <= 0.0002)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / tan((((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+17) || !((Math.PI * l) <= 0.0002)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / Math.tan((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+17) or not ((math.pi * l) <= 0.0002):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / math.tan((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+17) || !(Float64(pi * l) <= 0.0002))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / tan(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+17) || ~(((pi * l) <= 0.0002)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / tan((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[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 0.0002]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e17 or 2.0000000000000001e-4 < (*.f64 (PI.f64) l)

   1. Initial program 35.3

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

   2. Taylor expanded in l around 0 49.53

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

   3. Simplified 49.53

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

      [Start] 49.53	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      associate-/l* [=>] 49.53	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]
      unpow2 [=>] 49.53	\[ \pi \cdot \ell - \frac{\ell}{\frac{\color{blue}{F \cdot F}}{\pi}} \]
      associate-/r/ [=>] 49.53	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{F \cdot F} \cdot \pi} \]

   4. Taylor expanded in F around inf 1.25

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

   if -1e17 < (*.f64 (PI.f64) l) < 2.0000000000000001e-4

   1. Initial program 14.07

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

   2. Applied egg-rr 0.85

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.06

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 0.0002\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.04%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 0.0002\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2
Error	1.32%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 0.0002\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \end{array} \]

Alternative 3
Error	20.41%
Cost	7888

\[\begin{array}{l} t_0 := \frac{\pi \cdot \left(-\ell\right)}{F \cdot F}\\ t_1 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 2.8 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 1.6 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2.6 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Error	21.36%
Cost	7378

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-108} \lor \neg \left(\ell \leq -4.7 \cdot 10^{-191} \lor \neg \left(\ell \leq 1.6 \cdot 10^{-228}\right) \land \ell \leq 1.8 \cdot 10^{-154}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{-\pi}{F}\\ \end{array} \]

Alternative 5
Error	7.17%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -850000000000 \lor \neg \left(\ell \leq 5.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\ell}{F \cdot F}\right)\\ \end{array} \]

Alternative 6
Error	1.28%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1850000000000 \lor \neg \left(\ell \leq 5.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

Alternative 7
Error	20.57%
Cost	6528

\[\pi \cdot \ell \]

Alternative 8
Error	96.78%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))