VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 12.2

Time: 10.0s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

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) - ((1.0 / F) / (F / Math.tan((Math.PI * l))));
}

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) - ((1.0 / F) / (F / math.tan((math.pi * l))))

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(1.0 / F) / Float64(F / tan(Float64(pi * l)))))
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) - ((1.0 / F) / (F / tan((pi * l))));
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[(1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus F

Bits error versus l

Derivation

1. Initial program 16.4

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

2. Simplified 16.1

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

3. Applied egg-rr 12.2

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

4. Applied egg-rr 12.2

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

5. Final simplification 12.2

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

Reproduce

herbie shell --seed 2022133 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))