Average Error: 0.0 → 0.0
Time: 7.4s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(v \cdot v, -3, 1\right)\\ \mathsf{fma}\left(\sqrt[3]{t_0}, \sqrt[3]{\sqrt{t_0}} \cdot \sqrt{0.125}, \left(v \cdot v\right) \cdot \left(-\sqrt{t_0 \cdot 0.125}\right)\right) \end{array} \]
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (let* ((t_0 (fma (* v v) -3.0 1.0)))
   (fma
    (cbrt t_0)
    (* (cbrt (sqrt t_0)) (sqrt 0.125))
    (* (* v v) (- (sqrt (* t_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) {
	double t_0 = fma((v * v), -3.0, 1.0);
	return fma(cbrt(t_0), (cbrt(sqrt(t_0)) * sqrt(0.125)), ((v * v) * -sqrt((t_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)
	t_0 = fma(Float64(v * v), -3.0, 1.0)
	return fma(cbrt(t_0), Float64(cbrt(sqrt(t_0)) * sqrt(0.125)), Float64(Float64(v * v) * Float64(-sqrt(Float64(t_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_] := Block[{t$95$0 = N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision]}, N[(N[Power[t$95$0, 1/3], $MachinePrecision] * N[(N[Power[N[Sqrt[t$95$0], $MachinePrecision], 1/3], $MachinePrecision] * N[Sqrt[0.125], $MachinePrecision]), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * (-N[Sqrt[N[(t$95$0 * 0.125), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(v \cdot v, -3, 1\right)\\
\mathsf{fma}\left(\sqrt[3]{t_0}, \sqrt[3]{\sqrt{t_0}} \cdot \sqrt{0.125}, \left(v \cdot v\right) \cdot \left(-\sqrt{t_0 \cdot 0.125}\right)\right)
\end{array}

Error

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. Applied egg-rr0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022210 
(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))))