?

Average Error: 62.0 → 0
Time: 3.2s
Precision: binary64
Cost: 26432

?

\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
↓
\[\sqrt{{\left(\mathsf{fma}\left(y \cdot y, 2 - y \cdot y, 9 \cdot {x}^{4}\right)\right)}^{2}} \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
↓
(FPCore (x y)
 :precision binary64
 (sqrt (pow (fma (* y y) (- 2.0 (* y y)) (* 9.0 (pow x 4.0))) 2.0)))
double code(double x, double y) {
	return (9.0 * pow(x, 4.0)) - ((y * y) * ((y * y) - 2.0));
}
↓
double code(double x, double y) {
	return sqrt(pow(fma((y * y), (2.0 - (y * y)), (9.0 * pow(x, 4.0))), 2.0));
}

function code(x, y)
	return Float64(Float64(9.0 * (x ^ 4.0)) - Float64(Float64(y * y) * Float64(Float64(y * y) - 2.0)))
end
↓
function code(x, y)
	return sqrt((fma(Float64(y * y), Float64(2.0 - Float64(y * y)), Float64(9.0 * (x ^ 4.0))) ^ 2.0))
end

code[x_, y_] := N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[Sqrt[N[Power[N[(N[(y * y), $MachinePrecision] * N[(2.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]

9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\sqrt{{\left(\mathsf{fma}\left(y \cdot y, 2 - y \cdot y, 9 \cdot {x}^{4}\right)\right)}^{2}}

Error?

Derivation?

  1. Initial program 62.0

    \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]

  2. Applied egg-rr62.0

    \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(9, {x}^{4}, \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right)\right)\right)}^{2}}} \]

  3. Simplified0

    \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(y \cdot y, 2 - y \cdot y, 9 \cdot {x}^{4}\right)\right)}^{2}}} \]
    Proof

    [Start]62.0

    \[ \sqrt{{\left(\mathsf{fma}\left(9, {x}^{4}, \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right)\right)\right)}^{2}} \]

    fma-def [<=]62.0

    \[ \sqrt{{\color{blue}{\left(9 \cdot {x}^{4} + \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right)\right)}}^{2}} \]

    +-commutative [<=]62.0

    \[ \sqrt{{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(2 - y \cdot y\right) + 9 \cdot {x}^{4}\right)}}^{2}} \]

    fma-def [=>]0

    \[ \sqrt{{\color{blue}{\left(\mathsf{fma}\left(y \cdot y, 2 - y \cdot y, 9 \cdot {x}^{4}\right)\right)}}^{2}} \]

  4. Final simplification0

    \[\leadsto \sqrt{{\left(\mathsf{fma}\left(y \cdot y, 2 - y \cdot y, 9 \cdot {x}^{4}\right)\right)}^{2}} \]

Alternatives

Alternative 1
Error52.0
Cost7552
\[\left(9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) + \left(y \cdot y\right) \cdot 2 \]
Alternative 2
Error57.8
Cost6656
\[9 \cdot {x}^{4} \]
Alternative 3
Error57.9
Cost1600
\[\begin{array}{l} t_0 := y \cdot y + 3 \cdot \left(x \cdot x\right)\\ \left(y \cdot y\right) \cdot 2 + t_0 \cdot t_0 \end{array} \]

Error

Reproduce?

herbie shell --seed 2023053 
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))