Average Error: 0.0 → 0.0
Time: 1.9s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot \left({\left(\mathsf{fma}\left(v, v, -1\right)\right)}^{2} \cdot 0.125\right)}\right)\right) \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (expm1
  (log1p
   (sqrt (* (fma v (* v -3.0) 1.0) (* (pow (fma v v -1.0) 2.0) 0.125))))))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt((1.0 - (3.0 * (v * v))))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return expm1(log1p(sqrt((fma(v, (v * -3.0), 1.0) * (pow(fma(v, v, -1.0), 2.0) * 0.125)))));
}

function code(v)
	return Float64(Float64(Float64(sqrt(2.0) / 4.0) * sqrt(Float64(1.0 - Float64(3.0 * Float64(v * v))))) * Float64(1.0 - Float64(v * v)))
end
↓
function code(v)
	return expm1(log1p(sqrt(Float64(fma(v, Float64(v * -3.0), 1.0) * Float64((fma(v, v, -1.0) ^ 2.0) * 0.125)))))
end

code[v_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / 4.0), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(3.0 * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[v_] := N[(Exp[N[Log[1 + N[Sqrt[N[(N[(v * N[(v * -3.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Power[N[(v * v + -1.0), $MachinePrecision], 2.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot \left({\left(\mathsf{fma}\left(v, v, -1\right)\right)}^{2} \cdot 0.125\right)}\right)\right)

Error

Bits error versus v

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(\sqrt{2} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)} \]

  3. Applied egg-rr0.0

    \[\leadsto \sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \color{blue}{\sqrt{2 \cdot \left({\left(\mathsf{fma}\left(v, v, -1\right)\right)}^{2} \cdot 0.0625\right)}} \]

  4. Applied egg-rr0.0

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot \left({\left(\mathsf{fma}\left(v, v, -1\right)\right)}^{2} \cdot 0.125\right)}\right)\right)} \]

  5. Final simplification0.0

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right) \cdot \left({\left(\mathsf{fma}\left(v, v, -1\right)\right)}^{2} \cdot 0.125\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022155 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))