Average Error: 5.9 → 0.1
Time: 6.9s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\mathsf{fma}\left(y \cdot t, 4, \mathsf{fma}\left(x, x, \left(z \cdot \left(y \cdot z\right)\right) \cdot -4\right)\right) \]
(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma (* y t) 4.0 (fma x x (* (* z (* y z)) -4.0))))
double code(double x, double y, double z, double t) {
	return (x * x) - ((y * 4.0) * ((z * z) - t));
}
↓
double code(double x, double y, double z, double t) {
	return fma((y * t), 4.0, fma(x, x, ((z * (y * z)) * -4.0)));
}

function code(x, y, z, t)
	return Float64(Float64(x * x) - Float64(Float64(y * 4.0) * Float64(Float64(z * z) - t)))
end
↓
function code(x, y, z, t)
	return fma(Float64(y * t), 4.0, fma(x, x, Float64(Float64(z * Float64(y * z)) * -4.0)))
end

code[x_, y_, z_, t_] := N[(N[(x * x), $MachinePrecision] - N[(N[(y * 4.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[(y * t), $MachinePrecision] * 4.0 + N[(x * x + N[(N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(y \cdot t, 4, \mathsf{fma}\left(x, x, \left(z \cdot \left(y \cdot z\right)\right) \cdot -4\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.8
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Initial program 5.9

    \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]

  2. Simplified5.9

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

  3. Taylor expanded in y around 0 5.8

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

  4. Applied egg-rr6.0

    \[\leadsto \left(4 \cdot \left(y \cdot t\right) + {x}^{2}\right) - 4 \cdot \color{blue}{{\left(\sqrt[3]{y \cdot \left(z \cdot z\right)}\right)}^{3}} \]

  5. Applied egg-rr0.1

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

  6. Applied egg-rr0.1

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

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y \cdot t, 4, \mathsf{fma}\left(x, x, \left(z \cdot \left(y \cdot z\right)\right) \cdot -4\right)\right) \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4.0 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4.0) (- (* z z) t))))