Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 1.0

Time: 5.7s

Precision: binary64

Cost: 13952

Math

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

↓

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

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
 (/ 4.0 (* (* PI (- 3.0 (* 3.0 (* v v)))) (sqrt (- 2.0 (* v (* 6.0 v)))))))

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 4.0 / ((((double) M_PI) * (3.0 - (3.0 * (v * v)))) * sqrt((2.0 - (v * (6.0 * v)))));
}

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 4.0 / ((Math.PI * (3.0 - (3.0 * (v * v)))) * Math.sqrt((2.0 - (v * (6.0 * v)))));
}

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 4.0 / ((math.pi * (3.0 - (3.0 * (v * v)))) * math.sqrt((2.0 - (v * (6.0 * v)))))

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(4.0 / Float64(Float64(pi * Float64(3.0 - Float64(3.0 * Float64(v * v)))) * sqrt(Float64(2.0 - Float64(v * Float64(6.0 * v))))))
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 = 4.0 / ((pi * (3.0 - (3.0 * (v * v)))) * sqrt((2.0 - (v * (6.0 * v)))));
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[(4.0 / N[(N[(Pi * N[(3.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 - N[(v * N[(6.0 * v), $MachinePrecision]), $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_best_oopsla_all_46_json_45_simplify-7 [=>] 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 1.0

   \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right) + 0\right)} \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]

4. Simplified 1.0

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

   [Start] 1.0	\[ \frac{4}{\left(3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right) + 0\right) \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 1.0	\[ \frac{4}{\color{blue}{\left(3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)\right)} \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 1.0	\[ \frac{4}{\color{blue}{\left(\pi \cdot \left(3 \cdot \left(1 - v \cdot v\right)\right)\right)} \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-13 [=>] 1.0	\[ \frac{4}{\left(\pi \cdot \color{blue}{\left(1 \cdot 3 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]
   metadata-eval [=>] 1.0	\[ \frac{4}{\left(\pi \cdot \left(\color{blue}{3} - 3 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{2 - v \cdot \left(6 \cdot v\right)}} \]

5. Final simplification 1.0

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

Alternatives

Alternative 1
Error	1.0
Cost	13952

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

Alternative 2
Error	0.6
Cost	13184

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

Alternative 3
Error	1.6
Cost	13056

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

Reproduce

herbie shell --seed 2023090 
(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)))))))