VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.9%

Time: 15.0s

Precision: binary64

Cost: 32969

• could not determine a ground truth for program body

  1. F = 4.3781226391996485e-63
  2. l = 290603732654487.5

• could not determine a ground truth for program body

  1. F = -1.1779244528710493e+208
  2. l = -989065350799236.5

• could not determine a ground truth for program body

  1. F = -1.4741481103779622e-100
  2. l = 2880418639486607.5

• could not determine a ground truth for program body

  1. F = 4.68333820940088e+78
  2. l = -2054502901226846.5

• could not determine a ground truth for program body

  1. F = 4.075434013897819e-188
  2. l = -884455899619669.5

• could not determine a ground truth for program body

  1. F = 6.193188276212941e-56
  2. l = 940110349246116.5

• could not determine a ground truth for program body

  1. F = 6.737992123083833e-266
  2. l = -3331653601403693.5

• could not determine a ground truth for program body

  1. F = 5.6856814632754e-270
  2. l = 2571642173790166.5

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+20} \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+20) (not (<= (* PI l) 50000000000.0)))
   (* 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) <= -5e+20) || !((((double) M_PI) * l) <= 50000000000.0)) {
		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) <= -5e+20) || !((Math.PI * l) <= 50000000000.0)) {
		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) <= -5e+20) or not ((math.pi * l) <= 50000000000.0):
		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) <= -5e+20) || !(Float64(pi * l) <= 50000000000.0))
		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) <= -5e+20) || ~(((pi * l) <= 50000000000.0)))
		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], -5e+20], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 50000000000.0]], $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) < -5e20 or 5e10 < (*.f64 (PI.f64) l)

   1. Initial program 57.4%

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

   2. Simplified 57.4%

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

      [Start] 57.4	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 57.4	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 57.4	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around 0 47.0%

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

   4. Simplified 47.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)} \]
      Step-by-step derivation

      [Start] 47.0	\[ \ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right) \]
      unpow2 [=>] 47.0	\[ \ell \cdot \left(\pi - \frac{\pi}{\color{blue}{F \cdot F}}\right) \]

   5. Taylor expanded in F around inf 99.6%

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

   if -5e20 < (*.f64 (PI.f64) l) < 5e10

   1. Initial program 94.0%

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

   2. Applied egg-rr 99.0%

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

      [Start] 94.0	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 94.0	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-un-lft-identity [<=] 94.0	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]
      associate-/r* [=>] 99.0	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+20} \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.6%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000000000000 \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 2
Accuracy	98.3%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000000000000 \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000000000000 \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 4
Accuracy	92.3%
Cost	26441

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -10000000000000 \lor \neg \left(\pi \cdot \ell \leq 50000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-316} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-118}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \end{array} \]

Alternative 6
Accuracy	72.8%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-316} \lor \neg \left(F \cdot F \leq 2 \cdot 10^{-118}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{-\ell}{F \cdot \frac{F}{\pi}}\\ \end{array} \]

Alternative 7
Accuracy	73.0%
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023162 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))