Average Error: 1.0 → 0.0
Time: 8.8s
Precision: binary64
Cost: 20224
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\frac{\frac{\frac{1}{\pi}}{1 - v \cdot v} \cdot 1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
(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 (fma v (* v -6.0) 2.0))))
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(fma(v, (v * -6.0), 2.0));
}

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(fma(v, Float64(v * -6.0), 2.0)))
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[(N[(1.0 / Pi), $MachinePrecision] / N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] / N[Sqrt[N[(v * N[(v * -6.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $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}{\pi}}{1 - v \cdot v} \cdot 1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}

Error

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. Applied egg-rr1.0

    \[\leadsto \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(v \cdot v, -6, 2\right)\right)}^{1.5}}}} \]

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost20096
\[\frac{\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
Alternative 2
Error1.0
Cost13952
\[\frac{4}{\left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot 3\right)\right) \cdot \sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]
Alternative 3
Error0.7
Cost13504
\[\frac{1.3333333333333333}{\pi \cdot \left(1 - v \cdot v\right)} \cdot {2}^{-0.5} \]
Alternative 4
Error0.7
Cost13056
\[\frac{1.3333333333333333}{\pi} \cdot \sqrt{0.5} \]

Error

Reproduce

herbie shell --seed 2022294 
(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)))))))