VandenBroeck and Keller, Equation (6)

Average Error: 17.2 → 13.0

Time: 7.5s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (+ (* PI l) (* (pow (/ F (tan (* PI l))) -1.0) (/ -1.0 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) {
	return (((double) M_PI) * l) + (pow((F / tan((((double) M_PI) * l))), -1.0) * (-1.0 / F));
}

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) {
	return (Math.PI * l) + (Math.pow((F / Math.tan((Math.PI * l))), -1.0) * (-1.0 / F));
}

Python

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

↓

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

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)
	return Float64(Float64(pi * l) + Float64((Float64(F / tan(Float64(pi * l))) ^ -1.0) * Float64(-1.0 / F)))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

↓

function tmp = code(F, l)
	tmp = (pi * l) + (((F / tan((pi * l))) ^ -1.0) * (-1.0 / F));
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_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[Power[N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus F

Bits error versus l

Derivation

1. Initial program 17.2

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

2. Simplified 16.9

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

3. Applied egg-rr 13.0

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

4. Applied egg-rr 13.0

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

5. Final simplification 13.0

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

Reproduce

herbie shell --seed 2022150 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))