Average Error: 62.0 → 0
Time: 4.5s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
↓
\[\mathsf{fma}\left(y, y \cdot 2, -\mathsf{fma}\left(y \cdot y, y \cdot y, {x}^{4} \cdot -9\right)\right) \]
(FPCore (x y)
 :precision binary64
 (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))
↓
(FPCore (x y)
 :precision binary64
 (fma y (* y 2.0) (- (fma (* y y) (* y y) (* (pow x 4.0) -9.0)))))
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 fma(y, (y * 2.0), -fma((y * y), (y * y), (pow(x, 4.0) * -9.0)));
}

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

9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left(y, y \cdot 2, -\mathsf{fma}\left(y \cdot y, y \cdot y, {x}^{4} \cdot -9\right)\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. Simplified52.0

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

  3. Applied egg-rr52.0

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

  4. Applied egg-rr52.0

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

  5. Applied egg-rr52.0

    \[\leadsto \mathsf{fma}\left(y, y \cdot 2, -{\color{blue}{\left({\left(e^{{\log \left(\mathsf{fma}\left({x}^{4}, -9, {y}^{4}\right)\right)}^{0.25}}\right)}^{\left({\log \left(\mathsf{fma}\left({x}^{4}, -9, {y}^{4}\right)\right)}^{0.25}\right)}\right)}}^{\left(\sqrt{\log \left(\mathsf{fma}\left({x}^{4}, -9, {y}^{4}\right)\right)}\right)}\right) \]

  6. Applied egg-rr0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2022192 
(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))))