Average Error: 6.0 → 4.0
Time: 22.1s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 5.262792164914477010314770127344285061599 \cdot 10^{261}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 5.262792164914477010314770127344285061599 \cdot 10^{261}:\\
\;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\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}
double f(double x, double y, double z, double t) {
        double r366066 = x;
        double r366067 = r366066 * r366066;
        double r366068 = y;
        double r366069 = 4.0;
        double r366070 = r366068 * r366069;
        double r366071 = z;
        double r366072 = r366071 * r366071;
        double r366073 = t;
        double r366074 = r366072 - r366073;
        double r366075 = r366070 * r366074;
        double r366076 = r366067 - r366075;
        return r366076;
}
↓
double f(double x, double y, double z, double t) {
        double r366077 = z;
        double r366078 = r366077 * r366077;
        double r366079 = 5.262792164914477e+261;
        bool r366080 = r366078 <= r366079;
        double r366081 = x;
        double r366082 = r366081 * r366081;
        double r366083 = y;
        double r366084 = 4.0;
        double r366085 = r366083 * r366084;
        double r366086 = t;
        double r366087 = r366078 - r366086;
        double r366088 = r366085 * r366087;
        double r366089 = r366082 - r366088;
        double r366090 = sqrt(r366086);
        double r366091 = r366077 + r366090;
        double r366092 = r366085 * r366091;
        double r366093 = r366077 - r366090;
        double r366094 = r366092 * r366093;
        double r366095 = r366082 - r366094;
        double r366096 = r366080 ? r366089 : r366095;
        return r366096;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.0
Target6.0
Herbie4.0
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 5.262792164914477e+261

    1. Initial program 0.1

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

    if 5.262792164914477e+261 < (* z z)

    1. Initial program 48.0

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

    3. Applied add-sqr-sqrt56.8

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

    4. Applied difference-of-squares56.8

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

    5. Applied associate-*r*32.3

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 5.262792164914477010314770127344285061599 \cdot 10^{261}:\\ \;\;\;\;x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x - \left(\left(y \cdot 4\right) \cdot \left(z + \sqrt{t}\right)\right) \cdot \left(z - \sqrt{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))