Average Error: 6.3 → 0.1
Time: 5.4s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\mathsf{fma}\left(x, x, -\mathsf{fma}\left(y \cdot \left(4 \cdot z\right), z, -4 \cdot \left(y \cdot t\right)\right)\right) \]
(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t)
 :precision binary64
 (fma x x (- (fma (* y (* 4.0 z)) z (* -4.0 (* y t))))))
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(x, x, -fma((y * (4.0 * z)), z, (-4.0 * (y * t))));
}

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(x, x, Float64(-fma(Float64(y * Float64(4.0 * z)), z, Float64(-4.0 * Float64(y * t)))))
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[(x * x + (-N[(N[(y * N[(4.0 * z), $MachinePrecision]), $MachinePrecision] * z + N[(-4.0 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.3
Target6.3
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Initial program 6.3

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

  2. Taylor expanded in z around 0 6.3

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Taylor expanded in y around 0 0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022162 
(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))))