Average Error: 6.0 → 6.0
Time: 8.8s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\mathsf{fma}\left(y, 4 \cdot \left(t - z \cdot z\right), x \cdot x\right) \]
(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t) :precision binary64 (fma y (* 4.0 (- t (* z z))) (* x x)))
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, (4.0 * (t - (z * z))), (x * x));
}

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.0
Target6.0
Herbie6.0
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Initial program 6.0

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

  2. Simplified6.0

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

  3. Final simplification6.0

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

Reproduce

herbie shell --seed 2022131 
(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))))