Average Error: 6.2 → 0.1
Time: 4.5s
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 r615403 = x;
        double r615404 = r615403 * r615403;
        double r615405 = y;
        double r615406 = 4.0;
        double r615407 = r615405 * r615406;
        double r615408 = z;
        double r615409 = r615408 * r615408;
        double r615410 = t;
        double r615411 = r615409 - r615410;
        double r615412 = r615407 * r615411;
        double r615413 = r615404 - r615412;
        return r615413;
}
↓
double f(double x, double y, double z, double t) {
        double r615414 = x;
        double r615415 = y;
        double r615416 = 4.0;
        double r615417 = r615415 * r615416;
        double r615418 = t;
        double r615419 = r615417 * r615418;
        double r615420 = z;
        double r615421 = r615417 * r615420;
        double r615422 = -r615420;
        double r615423 = r615421 * r615422;
        double r615424 = r615419 + r615423;
        double r615425 = fma(r615414, r615414, r615424);
        return r615425;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.2
Target6.2
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.2

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

  2. Simplified6.2

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

  4. Applied sub-neg6.2

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

  5. Applied distribute-lft-in6.2

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

  7. Applied distribute-rgt-neg-in6.2

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

  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 2020062 +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))))