VandenBroeck and Keller, Equation (6)

Average Accuracy: 76.5% → 87.5%

Time: 15.6s

Precision: binary64

Cost: 46217

• could not determine a ground truth for program body

  1. F = 1.49564848843695e+53
  2. l = -2789954713270598.5

• could not determine a ground truth for program body

  1. F = -3.5412431031718533e+280
  2. l = -4049241298403972.5

Math

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

↓

\[\begin{array}{l} t_0 := \tan \left(\pi \cdot \ell\right)\\ \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{+28}\right):\\ \;\;\;\;\pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{F \cdot F} - \frac{0.5}{{F}^{4}}\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F \cdot \frac{F}{t_0}}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (tan (* PI l))))
   (if (or (<= (* PI l) -2e+16) (not (<= (* PI l) 5e+28)))
     (- (* PI l) (* (expm1 (- (/ 1.0 (* F F)) (/ 0.5 (pow F 4.0)))) t_0))
     (+ (* PI l) (/ -1.0 (* F (/ F t_0)))))))

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 t_0 = tan((((double) M_PI) * l));
	double tmp;
	if (((((double) M_PI) * l) <= -2e+16) || !((((double) M_PI) * l) <= 5e+28)) {
		tmp = (((double) M_PI) * l) - (expm1(((1.0 / (F * F)) - (0.5 / pow(F, 4.0)))) * t_0);
	} else {
		tmp = (((double) M_PI) * l) + (-1.0 / (F * (F / t_0)));
	}
	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 t_0 = Math.tan((Math.PI * l));
	double tmp;
	if (((Math.PI * l) <= -2e+16) || !((Math.PI * l) <= 5e+28)) {
		tmp = (Math.PI * l) - (Math.expm1(((1.0 / (F * F)) - (0.5 / Math.pow(F, 4.0)))) * t_0);
	} else {
		tmp = (Math.PI * l) + (-1.0 / (F * (F / t_0)));
	}
	return tmp;
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

↓

def code(F, l):
	t_0 = math.tan((math.pi * l))
	tmp = 0
	if ((math.pi * l) <= -2e+16) or not ((math.pi * l) <= 5e+28):
		tmp = (math.pi * l) - (math.expm1(((1.0 / (F * F)) - (0.5 / math.pow(F, 4.0)))) * t_0)
	else:
		tmp = (math.pi * l) + (-1.0 / (F * (F / t_0)))
	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)
	t_0 = tan(Float64(pi * l))
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+16) || !(Float64(pi * l) <= 5e+28))
		tmp = Float64(Float64(pi * l) - Float64(expm1(Float64(Float64(1.0 / Float64(F * F)) - Float64(0.5 / (F ^ 4.0)))) * t_0));
	else
		tmp = Float64(Float64(pi * l) + Float64(-1.0 / Float64(F * Float64(F / t_0))));
	end
	return 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_] := Block[{t$95$0 = N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e+28]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Exp[N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[Power[F, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(-1.0 / N[(F * N[(F / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e16 or 4.99999999999999957e28 < (*.f64 (PI.f64) l)

   1. Initial program 61.9%

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

   2. Applied egg-rr 61.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({F}^{-2}\right)\right)} \cdot \tan \left(\pi \cdot \ell\right) \]
      Step-by-step derivation

      [Start] 61.9	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      expm1-log1p-u [=>] 61.9	\[ \pi \cdot \ell - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{F \cdot F}\right)\right)} \cdot \tan \left(\pi \cdot \ell\right) \]
      pow2 [=>] 61.9	\[ \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\color{blue}{{F}^{2}}}\right)\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      pow-flip [=>] 61.9	\[ \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{F}^{\left(-2\right)}}\right)\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      metadata-eval [=>] 61.9	\[ \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left({F}^{\color{blue}{-2}}\right)\right) \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Taylor expanded in F around inf 77.7%

      \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\frac{1}{{F}^{2}} - 0.5 \cdot \frac{1}{{F}^{4}}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]

   4. Simplified 77.7%

      \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\frac{1}{F \cdot F} - \frac{0.5}{{F}^{4}}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      Step-by-step derivation

      [Start] 77.7	\[ \pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{{F}^{2}} - 0.5 \cdot \frac{1}{{F}^{4}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      unpow2 [=>] 77.7	\[ \pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{\color{blue}{F \cdot F}} - 0.5 \cdot \frac{1}{{F}^{4}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*r/ [=>] 77.7	\[ \pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{F \cdot F} - \color{blue}{\frac{0.5 \cdot 1}{{F}^{4}}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]
      metadata-eval [=>] 77.7	\[ \pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{F \cdot F} - \frac{\color{blue}{0.5}}{{F}^{4}}\right) \cdot \tan \left(\pi \cdot \ell\right) \]

   if -2e16 < (*.f64 (PI.f64) l) < 4.99999999999999957e28

   1. Initial program 85.0%

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

   2. Simplified 85.6%

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

      [Start] 85.0	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 85.6	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 85.6	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Applied egg-rr 98.6%

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

      [Start] 85.6	\[ \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F} \]
      associate-/r* [=>] 98.7	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
      div-inv [=>] 98.6	\[ \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}} \]

   4. Applied egg-rr 98.7%

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

      [Start] 98.6	\[ \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F} \]
      clear-num [=>] 98.6	\[ \pi \cdot \ell - \color{blue}{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}} \cdot \frac{1}{F} \]
      frac-times [=>] 98.7	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot 1}{\frac{F}{\tan \left(\pi \cdot \ell\right)} \cdot F}} \]
      metadata-eval [=>] 98.7	\[ \pi \cdot \ell - \frac{\color{blue}{1}}{\frac{F}{\tan \left(\pi \cdot \ell\right)} \cdot F} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{+28}\right):\\ \;\;\;\;\pi \cdot \ell - \mathsf{expm1}\left(\frac{1}{F \cdot F} - \frac{0.5}{{F}^{4}}\right) \cdot \tan \left(\pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{F \cdot \frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.1%
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -100000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	82.3%
Cost	19904

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

Alternative 3
Accuracy	82.3%
Cost	19776

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

Alternative 4
Accuracy	69.0%
Cost	13376

\[\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F} \]

Alternative 5
Accuracy	74.5%
Cost	13376

\[\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F} \]

Alternative 6
Accuracy	74.5%
Cost	13376

\[\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}} \]

Reproduce

herbie shell --seed 2023158 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))