Falkner and Boettcher, Equation (22+)

Average Accuracy: 98.5% → 100.0%

Time: 6.3s

Precision: binary64

Cost: 13952

Math

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

↓

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

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
 (/
  (* (/ (/ 1.0 PI) (- 1.0 (* v v))) 1.3333333333333333)
  (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 (((1.0 / ((double) M_PI)) / (1.0 - (v * v))) * 1.3333333333333333) / 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 (((1.0 / Math.PI) / (1.0 - (v * v))) * 1.3333333333333333) / 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 (((1.0 / math.pi) / (1.0 - (v * v))) * 1.3333333333333333) / 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(Float64(1.0 / pi) / Float64(1.0 - Float64(v * v))) * 1.3333333333333333) / sqrt(Float64(2.0 + Float64(Float64(v * 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 = (((1.0 / pi) / (1.0 - (v * v))) * 1.3333333333333333) / 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[(N[(1.0 / Pi), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(N[(v * v), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

2. Simplified 100.0%

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

   [Start] 98.5	\[ \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   associate-/r* [=>] 100.0	\[ \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]
   associate-*l* [=>] 100.0	\[ \frac{\frac{4}{\color{blue}{3 \cdot \left(\pi \cdot \left(1 - v \cdot v\right)\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   associate-/r* [=>] 100.0	\[ \frac{\color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   metadata-eval [=>] 100.0	\[ \frac{\frac{\color{blue}{1.3333333333333333}}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
   cancel-sign-sub-inv [=>] 100.0	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\color{blue}{2 + \left(-6\right) \cdot \left(v \cdot v\right)}}} \]
   *-commutative [=>] 100.0	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \color{blue}{\left(v \cdot v\right) \cdot \left(-6\right)}}} \]
   metadata-eval [=>] 100.0	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot \color{blue}{-6}}} \]

3. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
   div-inv [=>] 100.0	\[ \frac{\color{blue}{1.3333333333333333 \cdot \frac{1}{\pi \cdot \left(1 - v \cdot v\right)}}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
   *-commutative [=>] 100.0	\[ \frac{\color{blue}{\frac{1}{\pi \cdot \left(1 - v \cdot v\right)} \cdot 1.3333333333333333}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
   associate-/r* [=>] 100.0	\[ \frac{\color{blue}{\frac{\frac{1}{\pi}}{1 - v \cdot v}} \cdot 1.3333333333333333}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13824

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

Alternative 2
Accuracy	100.0%
Cost	13824

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

Alternative 3
Accuracy	99.0%
Cost	13568

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

Alternative 4
Accuracy	99.0%
Cost	13440

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

Alternative 5
Accuracy	99.0%
Cost	13056

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

Alternative 6
Accuracy	97.5%
Cost	12928

\[\frac{\sqrt{0.8888888888888888}}{\pi} \]

Reproduce

herbie shell --seed 2023140 
(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)))))))