Average Error: 1.0 → 0.0

Time: 18.3s

Precision: binary64

Cost: 26368

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

↓

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

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

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

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)) / sqrt(fma(Float64(v * v), -6.0, 2.0))) / pi)
end

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] / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

\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)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi}

Derivation

Initial program 1.0
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
Simplified0.0
\[\leadsto \color{blue}{\frac{1.3333333333333333}{\left(\pi \cdot \left(-\mathsf{fma}\left(v, v, -1\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]
Proof
Applied egg-rr0.0
\[\leadsto \color{blue}{\frac{\frac{\frac{-1.3333333333333333}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}}{\pi}} \]

Alternatives

Alternative 1
Error	0.0
Cost	20160

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

Alternative 2
Error	0.6
Cost	19712

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

Alternative 3
Error	1.0
Cost	13952

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

Reproduce

herbie shell --seed 2023010 
(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)))))))