Falkner and Boettcher, Equation (22+)

Average Accuracy: 98.5% → 100.0%

Time: 6.6s

Precision: binary64

Cost: 26368

Math

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

↓

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

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

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

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

   \[\leadsto \color{blue}{\frac{\frac{\frac{-1.3333333333333333}{\mathsf{fma}\left(v, v, -1\right)}}{\pi}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
   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	\[ \color{blue}{\frac{\frac{4}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}} \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13824

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

Alternative 2
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 3
Accuracy	98.9%
Cost	13568

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

Alternative 4
Accuracy	98.9%
Cost	13440

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

Alternative 5
Accuracy	98.9%
Cost	13056

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

Alternative 6
Accuracy	97.4%
Cost	12928

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

Reproduce

herbie shell --seed 2023138 
(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)))))))