Average Error: 5.9 → 0.1
Time: 4.1s
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 r698314 = x;
        double r698315 = r698314 * r698314;
        double r698316 = y;
        double r698317 = 4.0;
        double r698318 = r698316 * r698317;
        double r698319 = z;
        double r698320 = r698319 * r698319;
        double r698321 = t;
        double r698322 = r698320 - r698321;
        double r698323 = r698318 * r698322;
        double r698324 = r698315 - r698323;
        return r698324;
}
↓
double f(double x, double y, double z, double t) {
        double r698325 = x;
        double r698326 = y;
        double r698327 = 4.0;
        double r698328 = r698326 * r698327;
        double r698329 = t;
        double r698330 = r698328 * r698329;
        double r698331 = z;
        double r698332 = r698328 * r698331;
        double r698333 = -r698331;
        double r698334 = r698332 * r698333;
        double r698335 = r698330 + r698334;
        double r698336 = fma(r698325, r698325, r698335);
        return r698336;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target5.8
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. Simplified5.8

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

  4. Applied sub-neg5.8

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

  5. Applied distribute-lft-in5.8

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

  7. Applied distribute-rgt-neg-in5.8

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

  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 2020021 +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))))