Average Error: 6.0 → 3.0
Time: 31.1s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\
\;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r366049 = x;
        double r366050 = r366049 * r366049;
        double r366051 = y;
        double r366052 = 4.0;
        double r366053 = r366051 * r366052;
        double r366054 = z;
        double r366055 = r366054 * r366054;
        double r366056 = t;
        double r366057 = r366055 - r366056;
        double r366058 = r366053 * r366057;
        double r366059 = r366050 - r366058;
        return r366059;
}
↓
double f(double x, double y, double z, double t) {
        double r366060 = z;
        double r366061 = -1.74920595474598e+149;
        bool r366062 = r366060 <= r366061;
        double r366063 = 1.205875807953863e+154;
        bool r366064 = r366060 <= r366063;
        double r366065 = !r366064;
        bool r366066 = r366062 || r366065;
        double r366067 = x;
        double r366068 = r366067 * r366067;
        double r366069 = y;
        double r366070 = 4.0;
        double r366071 = r366069 * r366070;
        double r366072 = t;
        double r366073 = sqrt(r366072);
        double r366074 = r366060 + r366073;
        double r366075 = r366071 * r366074;
        double r366076 = r366060 - r366073;
        double r366077 = r366075 * r366076;
        double r366078 = r366068 - r366077;
        double r366079 = r366060 * r366060;
        double r366080 = r366072 - r366079;
        double r366081 = r366069 * r366080;
        double r366082 = r366070 * r366081;
        double r366083 = fma(r366067, r366067, r366082);
        double r366084 = r366066 ? r366078 : r366083;
        return r366084;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Split input into 2 regimes

  2. if z < -1.74920595474598e+149 or 1.205875807953863e+154 < z

    1. Initial program 61.6

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

    3. Applied add-sqr-sqrt62.6

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

    4. Applied difference-of-squares62.6

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

    5. Applied associate-*r*30.5

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

    if -1.74920595474598e+149 < z < 1.205875807953863e+154

    1. Initial program 0.1

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

    3. Applied fma-neg0.1

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

    4. Simplified0.1

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.74920595474597988684148347386440472749 \cdot 10^{149} \lor \neg \left(z \le 1.205875807953862938552183354802704820011 \cdot 10^{154}\right):\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot \left(t - z \cdot z\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))