VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 98.8%

Time: 14.7s

Precision: binary64

Cost: 33097

• could not determine a ground truth for program body

  1. F = -1.6222304372567625e-133
  2. l = 3530485164551244.5

• could not determine a ground truth for program body

  1. F = -1.0151333184728228e+224
  2. l = 3779289277188699.5

• could not determine a ground truth for program body

  1. F = 1.9172871609016836e+225
  2. l = -3291711771206441.5

• could not determine a ground truth for program body

  1. F = -8.021335114845761e+141
  2. l = 1428644307825117.5

• could not determine a ground truth for program body

  1. F = 5.4744586961933235e-101
  2. l = 2213528911522534.5

• could not determine a ground truth for program body

  1. F = -1.0843430393004452e-94
  2. l = 2152923106032281.5

Math

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

↓

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

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+16) (not (<= (* PI l) 2e-5)))
   (* PI l)
   (+ (* PI l) (/ (* (tan (* PI l)) (/ -1.0 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+16) || !((((double) M_PI) * l) <= 2e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((tan((((double) M_PI) * l)) * (-1.0 / 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+16) || !((Math.PI * l) <= 2e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((Math.tan((Math.PI * l)) * (-1.0 / 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+16) or not ((math.pi * l) <= 2e-5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((math.tan((math.pi * l)) * (-1.0 / 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+16) || !(Float64(pi * l) <= 2e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(tan(Float64(pi * l)) * Float64(-1.0 / 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+16) || ~(((pi * l) <= 2e-5)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((tan((pi * l)) * (-1.0 / 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+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e16 or 2.00000000000000016e-5 < (*.f64 (PI.f64) l)

   1. Initial program 62.0%

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

   2. Taylor expanded in l around 0 48.7%

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

   3. Simplified 48.7%

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

      [Start] 48.7	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      associate-/l* [=>] 48.7	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]
      associate-/r/ [=>] 48.7	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]
      unpow2 [=>] 48.7	\[ \pi \cdot \ell - \frac{\ell}{\color{blue}{F \cdot F}} \cdot \pi \]

   4. Taylor expanded in F around inf 99.5%

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

   if -5e16 < (*.f64 (PI.f64) l) < 2.00000000000000016e-5

   1. Initial program 91.1%

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

   2. Applied egg-rr 99.5%

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

      [Start] 91.1	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-/r* [=>] 91.2	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{1}{F}}{F}} \cdot \tan \left(\pi \cdot \ell\right) \]
      metadata-eval [<=] 91.2	\[ \pi \cdot \ell - \frac{\frac{\color{blue}{\sqrt{1}}}{F}}{F} \cdot \tan \left(\pi \cdot \ell\right) \]
      add-sqr-sqrt [=>] 45.9	\[ \pi \cdot \ell - \frac{\frac{\sqrt{1}}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}}{F} \cdot \tan \left(\pi \cdot \ell\right) \]
      sqrt-prod [<=] 70.1	\[ \pi \cdot \ell - \frac{\frac{\sqrt{1}}{\color{blue}{\sqrt{F \cdot F}}}}{F} \cdot \tan \left(\pi \cdot \ell\right) \]
      sqrt-div [<=] 70.1	\[ \pi \cdot \ell - \frac{\color{blue}{\sqrt{\frac{1}{F \cdot F}}}}{F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 70.1	\[ \pi \cdot \ell - \color{blue}{\frac{\sqrt{\frac{1}{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)}{F}} \]
      sqrt-div [=>] 70.3	\[ \pi \cdot \ell - \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{F \cdot F}}} \cdot \tan \left(\pi \cdot \ell\right)}{F} \]
      metadata-eval [=>] 70.3	\[ \pi \cdot \ell - \frac{\frac{\color{blue}{1}}{\sqrt{F \cdot F}} \cdot \tan \left(\pi \cdot \ell\right)}{F} \]
      sqrt-prod [=>] 49.4	\[ \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\sqrt{F} \cdot \sqrt{F}}} \cdot \tan \left(\pi \cdot \ell\right)}{F} \]
      add-sqr-sqrt [<=] 99.5	\[ \pi \cdot \ell - \frac{\frac{1}{\color{blue}{F}} \cdot \tan \left(\pi \cdot \ell\right)}{F} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	98.8%
Cost	32969

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

Alternative 2
Accuracy	98.5%
Cost	26569

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

Alternative 3
Accuracy	74.1%
Cost	7888

\[\begin{array}{l} t_0 := \pi \cdot \frac{-\ell}{F \cdot F}\\ t_1 := 1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{if}\;F \cdot F \leq 5 \cdot 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 6.9 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 5.3 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Accuracy	74.9%
Cost	7378

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{-18} \lor \neg \left(\ell \leq -3 \cdot 10^{-162}\right) \land \left(\ell \leq 6.2 \cdot 10^{-217} \lor \neg \left(\ell \leq 0.0065\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \]

Alternative 5
Accuracy	98.6%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1450000000000 \lor \neg \left(\ell \leq 115000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

Alternative 6
Accuracy	74.0%
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023161 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))