Average Error: 6.1 → 0.1
Time: 6.1s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z \cdot y, z \cdot -4, \mathsf{fma}\left(z \cdot y, z \cdot -4, z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, -4 \cdot \left(z \cdot z - t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(z \cdot y\right) \cdot \left(z \cdot -4\right)\\ \end{array} \]
(FPCore (x y z t) :precision binary64 (- (* x x) (* (* y 4.0) (- (* z z) t))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= z -1e+153)
   (fma
    x
    x
    (fma (* z y) (* z -4.0) (fma (* z y) (* z -4.0) (* z (* z (* y 4.0))))))
   (if (<= z 1e+140)
     (fma y (* -4.0 (- (* z z) t)) (* x x))
     (+ (* 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) {
	double tmp;
	if (z <= -1e+153) {
		tmp = fma(x, x, fma((z * y), (z * -4.0), fma((z * y), (z * -4.0), (z * (z * (y * 4.0))))));
	} else if (z <= 1e+140) {
		tmp = fma(y, (-4.0 * ((z * z) - t)), (x * x));
	} else {
		tmp = (x * x) + ((z * y) * (z * -4.0));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (z <= -1e+153)
		tmp = fma(x, x, fma(Float64(z * y), Float64(z * -4.0), fma(Float64(z * y), Float64(z * -4.0), Float64(z * Float64(z * Float64(y * 4.0))))));
	elseif (z <= 1e+140)
		tmp = fma(y, Float64(-4.0 * Float64(Float64(z * z) - t)), Float64(x * x));
	else
		tmp = Float64(Float64(x * x) + Float64(Float64(z * y) * Float64(z * -4.0)));
	end
	return tmp
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_] := If[LessEqual[z, -1e+153], N[(x * x + N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision] + N[(z * N[(z * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+140], N[(y * N[(-4.0 * N[(N[(z * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1 \cdot 10^{+153}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z \cdot y, z \cdot -4, \mathsf{fma}\left(z \cdot y, z \cdot -4, z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\right)\right)\right)\\

\mathbf{elif}\;z \leq 10^{+140}:\\
\;\;\;\;\mathsf{fma}\left(y, -4 \cdot \left(z \cdot z - t\right), x \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot x + \left(z \cdot y\right) \cdot \left(z \cdot -4\right)\\


\end{array}

Error

Target

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

Derivation

  1. Split input into 3 regimes

  2. if z < -1e153

    1. Initial program 62.5

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

    2. Taylor expanded in z around inf 62.5

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

    3. Simplified0.3

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

    4. Applied egg-rr0.3

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

    if -1e153 < z < 1.00000000000000006e140

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 1.00000000000000006e140 < z

    1. Initial program 54.1

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

    2. Taylor expanded in z around inf 54.4

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

    3. Simplified0.5

      \[\leadsto x \cdot x - \color{blue}{\left(z \cdot y\right) \cdot \left(4 \cdot z\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(z \cdot y, z \cdot -4, \mathsf{fma}\left(z \cdot y, z \cdot -4, z \cdot \left(z \cdot \left(y \cdot 4\right)\right)\right)\right)\right)\\ \mathbf{elif}\;z \leq 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, -4 \cdot \left(z \cdot z - t\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + \left(z \cdot y\right) \cdot \left(z \cdot -4\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))))