Average Error: 6.1 → 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 r682804 = x;
        double r682805 = r682804 * r682804;
        double r682806 = y;
        double r682807 = 4.0;
        double r682808 = r682806 * r682807;
        double r682809 = z;
        double r682810 = r682809 * r682809;
        double r682811 = t;
        double r682812 = r682810 - r682811;
        double r682813 = r682808 * r682812;
        double r682814 = r682805 - r682813;
        return r682814;
}
↓
double f(double x, double y, double z, double t) {
        double r682815 = x;
        double r682816 = y;
        double r682817 = 4.0;
        double r682818 = r682816 * r682817;
        double r682819 = t;
        double r682820 = r682818 * r682819;
        double r682821 = z;
        double r682822 = r682818 * r682821;
        double r682823 = -r682821;
        double r682824 = r682822 * r682823;
        double r682825 = r682820 + r682824;
        double r682826 = fma(r682815, r682815, r682825);
        return r682826;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.1
Target6.0
Herbie0.1
\[x \cdot x - 4 \cdot \left(y \cdot \left(z \cdot z - t\right)\right)\]

Derivation

  1. Initial program 6.1

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

  2. Simplified6.1

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

  4. Applied sub-neg6.1

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

  5. Applied distribute-lft-in6.1

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

  7. Applied distribute-rgt-neg-in6.1

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

  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 2020060 +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))))