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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 5.6499424326614904 \cdot 10^{293}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \cdot z \le 5.6499424326614904 \cdot 10^{293}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\

\end{array}

double f(double x, double y, double z, double t) {
        double r2965 = x;
        double r2966 = r2965 * r2965;
        double r2967 = y;
        double r2968 = 4.0;
        double r2969 = r2967 * r2968;
        double r2970 = z;
        double r2971 = r2970 * r2970;
        double r2972 = t;
        double r2973 = r2971 - r2972;
        double r2974 = r2969 * r2973;
        double r2975 = r2966 - r2974;
        return r2975;
}

↓

double f(double x, double y, double z, double t) {
        double r2976 = z;
        double r2977 = r2976 * r2976;
        double r2978 = 5.6499424326614904e+293;
        bool r2979 = r2977 <= r2978;
        double r2980 = x;
        double r2981 = y;
        double r2982 = 4.0;
        double r2983 = r2981 * r2982;
        double r2984 = t;
        double r2985 = r2984 - r2977;
        double r2986 = r2983 * r2985;
        double r2987 = fma(r2980, r2980, r2986);
        double r2988 = sqrt(r2984);
        double r2989 = r2988 + r2976;
        double r2990 = r2983 * r2989;
        double r2991 = r2988 - r2976;
        double r2992 = r2990 * r2991;
        double r2993 = fma(r2980, r2980, r2992);
        double r2994 = r2979 ? r2987 : r2993;
        return r2994;
}

Original	6.1
Target	6.1
Herbie	3.3

Derivation

Split input into 2 regimes
if (* z z) < 5.6499424326614904e+293
1. Initial program 0.1
  \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
2. Simplified0.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)}\]
if 5.6499424326614904e+293 < (* z z)
1. Initial program 59.7
  \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
2. Simplified59.7
  \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)}\]
3. Using strategy rm
4. Applied add-sqr-sqrt62.0
  \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(\color{blue}{\sqrt{t} \cdot \sqrt{t}} - z \cdot z\right)\right)\]
5. Applied difference-of-squares62.0
  \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \color{blue}{\left(\left(\sqrt{t} + z\right) \cdot \left(\sqrt{t} - z\right)\right)}\right)\]
6. Applied associate-*r*31.7
  \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)}\right)\]
Recombined 2 regimes into one program.
Final simplification3.3
\[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 5.6499424326614904 \cdot 10^{293}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Shading:$sradialGradientWithFocusShader from Rasterific-0.6.1, B"
  :precision binary64

  :herbie-target
  (- (* x x) (* 4 (* y (- (* z z) t))))

  (- (* x x) (* (* y 4) (- (* z z) t))))

Error

Target

Derivation

`if (* z z) < 5.6499424326614904e+293`

`if 5.6499424326614904e+293 < (* z z)`

Reproduce