VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 0.7

Time: 14.1s

Precision: binary64

Cost: 33096

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -4e+22)
   (* PI l)
   (if (<= (* PI l) 500000000.0)
     (+ (* PI l) (/ (/ -1.0 F) (/ F (tan (* PI l)))))
     (* 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) <= -4e+22) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 500000000.0) {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / tan((((double) M_PI) * l))));
	} else {
		tmp = ((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) <= -4e+22) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 500000000.0) {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / Math.tan((Math.PI * l))));
	} else {
		tmp = 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) <= -4e+22:
		tmp = math.pi * l
	elif (math.pi * l) <= 500000000.0:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / math.tan((math.pi * l))))
	else:
		tmp = 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) <= -4e+22)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 500000000.0)
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / tan(Float64(pi * l)))));
	else
		tmp = 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) <= -4e+22)
		tmp = pi * l;
	elseif ((pi * l) <= 500000000.0)
		tmp = (pi * l) + ((-1.0 / F) / (F / tan((pi * l))));
	else
		tmp = 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[LessEqual[N[(Pi * l), $MachinePrecision], -4e+22], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 500000000.0], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -4e22 or 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 23.0

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

   2. Simplified 23.0

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      Proof
      (-.f64 (*.f64 (PI.f64) l) (/.f64 (tan.f64 (*.f64 (PI.f64) l)) (*.f64 F F))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (PI.f64) l) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (tan.f64 (*.f64 (PI.f64) l)))) (*.f64 F F))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (PI.f64) l) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))): 13 points increase in error, 6 points decrease in error

   3. Taylor expanded in l around inf 0.5

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

   if -4e22 < (*.f64 (PI.f64) l) < 5e8

   1. Initial program 9.5

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.9

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

   4. Applied egg-rr 0.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternatives

Alternative 1
Error	14.3
Cost	46424

\[\begin{array}{l} t_0 := \frac{\pi}{F \cdot \frac{-F}{\ell}}\\ \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{-67}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F \cdot \left(-F\right)}\\ \mathbf{elif}\;\pi \cdot \ell \leq -1 \cdot 10^{-182}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\pi \cdot \ell \leq 10^{-270}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2
Error	0.7
Cost	32968

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Error	1.0
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell \cdot \frac{\pi}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Error	1.0
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	1.0
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6
Error	1.0
Cost	20104

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7
Error	13.6
Cost	7888

\[\begin{array}{l} t_0 := \pi \cdot \frac{-\ell}{F \cdot F}\\ t_1 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 10^{-318}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 4.238873562306936 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 5.717413497051987 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 1.0874890540105192 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8
Error	14.3
Cost	7640

\[\begin{array}{l} t_0 := -\frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-68}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-185}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.7694864841975086 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.052425611311771 \cdot 10^{-271}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 9
Error	14.3
Cost	7640

\[\begin{array}{l} t_0 := \frac{\pi}{F \cdot \frac{-F}{\ell}}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-68}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;\pi \cdot \frac{-\ell}{F \cdot F}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-185}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq -1.7694864841975086 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3.052425611311771 \cdot 10^{-271}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 10
Error	13.5
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2022306 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))