Average Error: 5.9 → 0.1
Time: 3.5s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot z\right) \cdot \left(-z\right)\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot z\right) \cdot \left(-z\right)\right)
double f(double x, double y, double z, double t) {
        double r605206 = x;
        double r605207 = r605206 * r605206;
        double r605208 = y;
        double r605209 = 4.0;
        double r605210 = r605208 * r605209;
        double r605211 = z;
        double r605212 = r605211 * r605211;
        double r605213 = t;
        double r605214 = r605212 - r605213;
        double r605215 = r605210 * r605214;
        double r605216 = r605207 - r605215;
        return r605216;
}
↓
double f(double x, double y, double z, double t) {
        double r605217 = x;
        double r605218 = y;
        double r605219 = 4.0;
        double r605220 = r605218 * r605219;
        double r605221 = t;
        double r605222 = r605220 * r605221;
        double r605223 = z;
        double r605224 = r605220 * r605223;
        double r605225 = -r605223;
        double r605226 = r605224 * r605225;
        double r605227 = r605222 + r605226;
        double r605228 = fma(r605217, r605217, r605227);
        return r605228;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.9
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 5.9

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

  2. Simplified5.9

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

  4. Applied sub-neg5.9

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

  5. Applied distribute-lft-in5.9

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

  7. Applied distribute-rgt-neg-in5.9

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020001 +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))))