Average Error: 6.3 → 3.4
Time: 8.5s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 7.0107442232151245 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array} \]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \leq 7.0107442232151245 \cdot 10^{+301}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\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 z) 7.0107442232151245e+301)
   (fma x x (* (* y 4.0) (- t (* z z))))
   (- (* x x) (* (* (* y 4.0) (+ z (sqrt t))) (- z (sqrt 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) {
	double tmp;
	if ((z * z) <= 7.0107442232151245e+301) {
		tmp = fma(x, x, ((y * 4.0) * (t - (z * z))));
	} else {
		tmp = (x * x) - (((y * 4.0) * (z + sqrt(t))) * (z - sqrt(t)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.3
Target6.3
Herbie3.4
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 z z) < 7.0107442232151245e301

    1. Initial program 0.1

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

    2. Applied fma-neg_binary640.1

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

    if 7.0107442232151245e301 < (*.f64 z z)

    1. Initial program 61.6

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

    2. Applied add-sqr-sqrt_binary6462.8

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

    3. Applied difference-of-squares_binary6462.8

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

    4. Applied associate-*r*_binary6433.3

      \[\leadsto x \cdot x - \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 7.0107442232151245 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))