Average Error: 0.0 → 0.0
Time: 7.2s
Precision: binary64
Cost: 14016
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\begin{array}{l} t_0 := 1 - v \cdot v\\ \sqrt{\left(t_0 \cdot t_0\right) \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125\right)} \end{array} \]
(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 (- 1.0 (* v v))))
   (sqrt (* (* t_0 t_0) (* (fma (* v v) -3.0 1.0) 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 = 1.0 - (v * v);
	return sqrt(((t_0 * t_0) * (fma((v * v), -3.0, 1.0) * 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 = Float64(1.0 - Float64(v * v))
	return sqrt(Float64(Float64(t_0 * t_0) * Float64(fma(Float64(v * v), -3.0, 1.0) * 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[(1.0 - N[(v * v), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[(N[(v * v), $MachinePrecision] * -3.0 + 1.0), $MachinePrecision] * 0.125), $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 := 1 - v \cdot v\\
\sqrt{\left(t_0 \cdot t_0\right) \cdot \left(\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125\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

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.0
Cost13632
\[\left(1 - v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right) \cdot 0.125} \]
Alternative 2
Error0.2
Cost7744
\[\left(1 - v \cdot v\right) \cdot \left(\left(1 + \left(v \cdot v\right) \cdot \left(-1.5 + \left(v \cdot v\right) \cdot -1.125\right)\right) \cdot \sqrt{0.125}\right) \]
Alternative 3
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.625\right) \]
Alternative 4
Error0.7
Cost6848
\[\left(1 - v \cdot v\right) \cdot \sqrt{0.125} \]
Alternative 5
Error0.7
Cost6464
\[\sqrt{0.125} \]

Error

Reproduce

herbie shell --seed 2022301 
(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))))