Average Error: 6.0 → 3.1
Time: 14.6s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.262760484415064360933171183565398999574 \cdot 10^{159}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \mathbf{elif}\;z \le 3.942332962792774130120686152893868508978 \cdot 10^{151}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - {z}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - {\left(\sqrt{z}\right)}^{2}\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.262760484415064360933171183565398999574 \cdot 10^{159}:\\
\;\;\;\;\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)\\

\mathbf{elif}\;z \le 3.942332962792774130120686152893868508978 \cdot 10^{151}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - {z}^{2}\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} - {\left(\sqrt{z}\right)}^{2}\right)\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r862211 = x;
        double r862212 = r862211 * r862211;
        double r862213 = y;
        double r862214 = 4.0;
        double r862215 = r862213 * r862214;
        double r862216 = z;
        double r862217 = r862216 * r862216;
        double r862218 = t;
        double r862219 = r862217 - r862218;
        double r862220 = r862215 * r862219;
        double r862221 = r862212 - r862220;
        return r862221;
}
↓
double f(double x, double y, double z, double t) {
        double r862222 = z;
        double r862223 = -3.2627604844150644e+159;
        bool r862224 = r862222 <= r862223;
        double r862225 = x;
        double r862226 = y;
        double r862227 = 4.0;
        double r862228 = r862226 * r862227;
        double r862229 = t;
        double r862230 = sqrt(r862229);
        double r862231 = r862230 + r862222;
        double r862232 = r862228 * r862231;
        double r862233 = r862230 - r862222;
        double r862234 = r862232 * r862233;
        double r862235 = fma(r862225, r862225, r862234);
        double r862236 = 3.942332962792774e+151;
        bool r862237 = r862222 <= r862236;
        double r862238 = 2.0;
        double r862239 = pow(r862222, r862238);
        double r862240 = r862229 - r862239;
        double r862241 = r862228 * r862240;
        double r862242 = fma(r862225, r862225, r862241);
        double r862243 = sqrt(r862222);
        double r862244 = pow(r862243, r862238);
        double r862245 = r862230 - r862244;
        double r862246 = r862232 * r862245;
        double r862247 = fma(r862225, r862225, r862246);
        double r862248 = r862237 ? r862242 : r862247;
        double r862249 = r862224 ? r862235 : r862248;
        return r862249;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.0
Target6.0
Herbie3.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.2627604844150644e+159

    1. Initial program 64.0

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied fma-neg64.0

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

    4. Simplified64.0

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(t - {z}^{2}\right)}\right)\]
    5. Using strategy rm

    6. Applied unpow264.0

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

    7. Applied add-sqr-sqrt64.0

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

    8. Applied difference-of-squares64.0

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

    9. Applied associate-*r*29.9

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

    if -3.2627604844150644e+159 < z < 3.942332962792774e+151

    1. Initial program 0.3

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied fma-neg0.3

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

    4. Simplified0.3

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

    if 3.942332962792774e+151 < z

    1. Initial program 61.9

      \[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
    2. Using strategy rm

    3. Applied fma-neg61.9

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

    4. Simplified61.9

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot 4\right) \cdot \left(t - {z}^{2}\right)}\right)\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt61.9

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

    7. Applied unpow-prod-down61.9

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

    8. Applied add-sqr-sqrt62.9

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

    9. Applied difference-of-squares62.9

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

    10. Applied associate-*r*32.6

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

    11. Simplified32.5

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.262760484415064360933171183565398999574 \cdot 10^{159}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - z\right)\right)\\ \mathbf{elif}\;z \le 3.942332962792774130120686152893868508978 \cdot 10^{151}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot \left(t - {z}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot \left(\sqrt{t} + z\right)\right) \cdot \left(\sqrt{t} - {\left(\sqrt{z}\right)}^{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))))