Average Error: 1.0 → 0.0
Time: 2.2s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\frac{-1.3333333333333333}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}\right)\right) \]
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
↓
(FPCore (v)
 :precision binary64
 (log1p
  (expm1
   (/
    (/ (/ -1.3333333333333333 PI) (fma v v -1.0))
    (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 log1p(expm1((((-1.3333333333333333 / ((double) M_PI)) / fma(v, v, -1.0)) / 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 log1p(expm1(Float64(Float64(Float64(-1.3333333333333333 / pi) / fma(v, v, -1.0)) / sqrt(fma(Float64(v * 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[Log[1 + N[(Exp[N[(N[(N[(-1.3333333333333333 / Pi), $MachinePrecision] / N[(v * v + -1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(v * v), $MachinePrecision] * -6.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $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)}}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\frac{-1.3333333333333333}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}\right)\right)

Error

Bits error versus v

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. Simplified0.0

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

  3. Applied egg-rr1.0

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

  4. Applied egg-rr0.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\frac{-1.3333333333333333}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}\right)\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{\frac{-1.3333333333333333}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}\right)\right) \]

Reproduce

herbie shell --seed 2022178 
(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)))))))