Average Error: 6.1 → 0.5
Time: 6.5s
Precision: binary64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.424915612600789 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot \left(t - z \cdot z\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)\\ \end{array} \]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \leq 1.424915612600789 \cdot 10^{+211}:\\
\;\;\;\;\mathsf{fma}\left(y, 4 \cdot \left(t - z \cdot z\right), x \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\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) 1.424915612600789e+211)
   (fma y (* 4.0 (- t (* z z))) (* x x))
   (fma z (* z (* y -4.0)) (* 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) {
	double tmp;
	if ((z * z) <= 1.424915612600789e+211) {
		tmp = fma(y, (4.0 * (t - (z * z))), (x * x));
	} else {
		tmp = fma(z, (z * (y * -4.0)), (x * x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.1
Target6.1
Herbie0.5
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 z z) < 1.42491561260078895e211

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if 1.42491561260078895e211 < (*.f64 z z)

    1. Initial program 36.8

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

    2. Simplified37.2

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

    3. Taylor expanded in y around 0 36.8

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

    4. Simplified36.8

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

    5. Taylor expanded in t around 0 39.1

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

    6. Simplified2.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.424915612600789 \cdot 10^{+211}:\\ \;\;\;\;\mathsf{fma}\left(y, 4 \cdot \left(t - z \cdot z\right), x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z \cdot \left(y \cdot -4\right), x \cdot x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022076 
(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))))