VandenBroeck and Keller, Equation (6)

Average Error: 16.5 → 0.7

Time: 19.4s

Precision: binary64

Cost: 32968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -2e+20)
   (* PI l)
   (if (<= (* PI l) 20000000000.0)
     (- (* PI l) (/ (/ (tan (* PI l)) F) F))
     (* 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) <= -2e+20) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 20000000000.0) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} 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) <= -2e+20) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 20000000000.0) {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	} 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) <= -2e+20:
		tmp = math.pi * l
	elif (math.pi * l) <= 20000000000.0:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	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) <= -2e+20)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 20000000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	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) <= -2e+20)
		tmp = pi * l;
	elseif ((pi * l) <= 20000000000.0)
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	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], -2e+20], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 20000000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e20 or 2e10 < (*.f64 (PI.f64) l)

   1. Initial program 23.2

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

   2. Simplified 23.2

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      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))))): 14 points increase in error, 3 points decrease in error

   3. Taylor expanded in l around 0 31.5

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

   4. Simplified 31.5

      \[\leadsto \pi \cdot \ell - \color{blue}{\pi \cdot \frac{\ell}{F \cdot F}} \]
      Proof
      (*.f64 (PI.f64) (/.f64 l (*.f64 F F))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (/.f64 l (Rewrite<= unpow2_binary64 (pow.f64 F 2)))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l (pow.f64 F 2)) (PI.f64))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 l (PI.f64)) (pow.f64 F 2))): 17 points increase in error, 21 points decrease in error

   5. Taylor expanded in F around inf 0.5

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

   if -2e20 < (*.f64 (PI.f64) l) < 2e10

   1. Initial program 9.6

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

   2. Applied egg-rr 1.0

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.377413575976211 \cdot 10^{+19}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 10200:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{\frac{F}{\ell}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 2
Error	13.9
Cost	7640

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ t_1 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -597962464045.8406:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -3.6370355880284046 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.92268577947211 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.015240138882216 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7640

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ t_1 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -597962464045.8406:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -3.6370355880284046 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.92268577947211 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.015240138882216 \cdot 10^{-12}:\\ \;\;\;\;\left(-\pi\right) \cdot \frac{\ell}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Error	1.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.377413575976211 \cdot 10^{+19}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 10200:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	13.6
Cost	6528

\[\pi \cdot \ell \]

Alternative 6
Error	61.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022313 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))