Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.0

Time: 6.6s

Precision: binary64

Cost: 13824

Math

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

↓

\[\frac{\frac{\frac{1.3333333333333333}{1 - v \cdot v}}{\pi}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

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 (- 1.0 (* v v))) PI)
  (sqrt (- 2.0 (* v (* v 6.0))))))

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 / (1.0 - (v * v))) / ((double) M_PI)) / sqrt((2.0 - (v * (v * 6.0))));
}

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 / (1.0 - (v * v))) / Math.PI) / Math.sqrt((2.0 - (v * (v * 6.0))));
}

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 / (1.0 - (v * v))) / math.pi) / math.sqrt((2.0 - (v * (v * 6.0))))

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(Float64(Float64(1.3333333333333333 / Float64(1.0 - Float64(v * v))) / pi) / sqrt(Float64(2.0 - Float64(v * Float64(v * 6.0)))))
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 / (1.0 - (v * v))) / pi) / sqrt((2.0 - (v * (v * 6.0))));
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[(N[(N[(1.3333333333333333 / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] / N[Sqrt[N[(2.0 - N[(v * N[(v * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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

   \[\leadsto \color{blue}{\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}}} \]
   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.json-simplify-50 [=>] 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.json-simplify-50 [<=] 1.0	\[ \frac{4}{\color{blue}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]
   rational.json-simplify-24 [=>] 1.0	\[ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - \color{blue}{v \cdot \left(6 \cdot v\right)}}} \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} + 0} \]

4. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{1.3333333333333333}{1 - v \cdot v}}{\pi}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}} \]
   Proof

   [Start] 0.0	\[ \frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} + 0 \]
   rational.json-simplify-8 [=>] 0.0	\[ \color{blue}{\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}}} \]
   rational.json-simplify-15 [=>] 0.0	\[ \frac{\color{blue}{\frac{\frac{1.3333333333333333}{1 - v \cdot v}}{\pi}}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

5. Final simplification 0.0

   \[\leadsto \frac{\frac{\frac{1.3333333333333333}{1 - v \cdot v}}{\pi}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

Alternatives

Alternative 1
Error	0.0
Cost	13824

\[\frac{\frac{\frac{1.3333333333333333}{\pi}}{1 - v \cdot v}}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

Alternative 2
Error	0.6
Cost	13568

\[\frac{\frac{1}{\pi} \cdot 1.3333333333333333}{\sqrt{2 - v \cdot \left(v \cdot 6\right)}} \]

Alternative 3
Error	0.6
Cost	13440

\[\frac{1.3333333333333333}{\pi \cdot \sqrt{2 + v \cdot \left(v \cdot -6\right)}} \]

Alternative 4
Error	1.6
Cost	13056

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

Alternative 5
Error	0.6
Cost	13056

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

Reproduce

herbie shell --seed 2023073 
(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)))))))