Average Error: 62.0 → 0
Time: 1.3s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
↓
\[y \cdot \left(y \cdot 2\right) - \mathsf{fma}\left(y \cdot y, y \cdot y, -9 \cdot {x}^{4}\right) \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
↓
(FPCore (x y)
 :precision binary64
 (- (* y (* y 2.0)) (fma (* y y) (* y y) (- (* 9.0 (pow x 4.0))))))
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 (y * (y * 2.0)) - fma((y * y), (y * y), -(9.0 * pow(x, 4.0)));
}

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 Float64(Float64(y * Float64(y * 2.0)) - fma(Float64(y * y), Float64(y * y), Float64(-Float64(9.0 * (x ^ 4.0)))))
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[(N[(y * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + (-N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 62.0

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

  2. Simplified62.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(9, {x}^{4}, y \cdot \left(y \cdot 2 - {y}^{3}\right)\right)} \]

  3. Applied egg-rr52.0

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

  4. Applied egg-rr0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2022155 
(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))))