VandenBroeck and Keller, Equation (6)

Average Accuracy: 74.3% → 97.7%

Time: 15.7s

Precision: binary64

Cost: 33097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-51}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \]

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+24) (not (<= (* PI l) 4e-51)))
   (* PI l)
   (+ (* PI l) (* (/ (tan (* PI l)) F) (/ -1.0 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+24) || !((((double) M_PI) * l) <= 4e-51)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) / F) * (-1.0 / 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) <= -1e+24) || !((Math.PI * l) <= 4e-51)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) / F) * (-1.0 / 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) <= -1e+24) or not ((math.pi * l) <= 4e-51):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) / F) * (-1.0 / 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+24) || !(Float64(pi * l) <= 4e-51))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) / F) * Float64(-1.0 / 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) <= -1e+24) || ~(((pi * l) <= 4e-51)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) / F) * (-1.0 / 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], -1e+24], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 4e-51]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -9.9999999999999998e23 or 4e-51 < (*.f64 (PI.f64) l)

   1. Initial program 65.8%

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

   2. Simplified 65.9%

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

      [Start] 65.8	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 65.9	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 65.9	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around inf 97.0%

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

   if -9.9999999999999998e23 < (*.f64 (PI.f64) l) < 4e-51

   1. Initial program 84.4%

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

   2. Simplified 85.3%

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

      [Start] 84.4	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 85.3	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 85.3	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Applied egg-rr 98.5%

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

      [Start] 85.3	\[ \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F} \]
      associate-/r* [=>] 98.6	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
      div-inv [=>] 98.5	\[ \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-51}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{-1}{F}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+24} \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-51}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5000000000 \lor \neg \left(\pi \cdot \ell \leq 4 \cdot 10^{-51}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 3
Accuracy	91.4%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1750000000 \lor \neg \left(\ell \leq 8.8 \cdot 10^{-69}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1750000000 \lor \neg \left(\ell \leq 1.7 \cdot 10^{-51}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 5
Accuracy	91.0%
Cost	13513

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1200000000 \lor \neg \left(\ell \leq 8.8 \cdot 10^{-69}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 6
Accuracy	80.1%
Cost	7888

\[\begin{array}{l} t_0 := -\pi \cdot \frac{\ell}{F \cdot F}\\ t_1 := 1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 1.7 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 4.5 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 4.5 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7
Accuracy	79.9%
Cost	7440

\[\begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{-9}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-182}:\\ \;\;\;\;\pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-183}:\\ \;\;\;\;1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-46}:\\ \;\;\;\;\pi \cdot \left(\frac{\ell}{F} \cdot \frac{-1}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8
Accuracy	80.1%
Cost	7376

\[\begin{array}{l} t_0 := \pi \cdot \left(-\frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{if}\;F \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-186}:\\ \;\;\;\;1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 9
Accuracy	79.4%
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023140 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))