Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.6

Time: 6.9s

Precision: binary64

Cost: 13056

Math

\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

↓

\[\frac{1.3333333333333333}{\sqrt{2} \cdot \pi} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

↓

(FPCore (v) :precision binary64 (/ 1.3333333333333333 (* (sqrt 2.0) PI)))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
}

↓

double code(double v) {
	return 1.3333333333333333 / (sqrt(2.0) * ((double) M_PI));
}

Java

public static double code(double v) {
	return 4.0 / (((3.0 * Math.PI) * (1.0 - (v * v))) * Math.sqrt((2.0 - (6.0 * (v * v)))));
}

↓

public static double code(double v) {
	return 1.3333333333333333 / (Math.sqrt(2.0) * Math.PI);
}

Python

def code(v):
	return 4.0 / (((3.0 * math.pi) * (1.0 - (v * v))) * math.sqrt((2.0 - (6.0 * (v * v)))))

↓

def code(v):
	return 1.3333333333333333 / (math.sqrt(2.0) * math.pi)

Julia

function code(v)
	return Float64(4.0 / Float64(Float64(Float64(3.0 * pi) * Float64(1.0 - Float64(v * v))) * sqrt(Float64(2.0 - Float64(6.0 * Float64(v * v))))))
end

↓

function code(v)
	return Float64(1.3333333333333333 / Float64(sqrt(2.0) * pi))
end

MATLAB

function tmp = code(v)
	tmp = 4.0 / (((3.0 * pi) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v)))));
end

↓

function tmp = code(v)
	tmp = 1.3333333333333333 / (sqrt(2.0) * pi);
end

Wolfram

code[v_] := N[(4.0 / N[(N[(N[(3.0 * Pi), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(6.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[v_] := N[(1.3333333333333333 / N[(N[Sqrt[2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.0

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

2. Simplified 1.0

   \[\leadsto \color{blue}{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(\sqrt{2 - v \cdot \left(v \cdot 6\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   Proof

   [Start] 1.0	\[ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   rational_best-simplify-2 [=>] 1.0	\[ \frac{4}{\color{blue}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right)}} \]
   rational_best-simplify-44 [=>] 1.0	\[ \frac{4}{\color{blue}{\left(3 \cdot \pi\right) \cdot \left(\sqrt{2 - 6 \cdot \left(v \cdot v\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]
   rational_best-simplify-44 [=>] 1.0	\[ \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\sqrt{2 - \color{blue}{v \cdot \left(6 \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]
   rational_best-simplify-2 [=>] 1.0	\[ \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\sqrt{2 - v \cdot \color{blue}{\left(v \cdot 6\right)}} \cdot \left(1 - v \cdot v\right)\right)} \]

3. Taylor expanded in v around 0 1.6

   \[\leadsto \frac{4}{\left(3 \cdot \pi\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot \left(1 - v \cdot v\right)\right)} \]

4. Taylor expanded in v around 0 0.6

   \[\leadsto \color{blue}{\frac{1.3333333333333333}{\sqrt{2} \cdot \pi}} \]

5. Final simplification 0.6

   \[\leadsto \frac{1.3333333333333333}{\sqrt{2} \cdot \pi} \]

Alternatives

Alternative 1
Error	1.6
Cost	13056

\[1.3333333333333333 \cdot \frac{\sqrt{0.5}}{\pi} \]

Reproduce

herbie shell --seed 2023096 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))