Average Error: 6.0 → 0.1
Time: 3.7s
Precision: 64
\[x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)\]
↓
\[\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \sqrt[3]{{0}^{\frac{3}{2}} \cdot {0}^{\frac{3}{2}}}\right)\]
x \cdot x - \left(y \cdot 4\right) \cdot \left(z \cdot z - t\right)
↓
\mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \sqrt[3]{{0}^{\frac{3}{2}} \cdot {0}^{\frac{3}{2}}}\right)
double f(double x, double y, double z, double t) {
        double r573241 = x;
        double r573242 = r573241 * r573241;
        double r573243 = y;
        double r573244 = 4.0;
        double r573245 = r573243 * r573244;
        double r573246 = z;
        double r573247 = r573246 * r573246;
        double r573248 = t;
        double r573249 = r573247 - r573248;
        double r573250 = r573245 * r573249;
        double r573251 = r573242 - r573250;
        return r573251;
}
↓
double f(double x, double y, double z, double t) {
        double r573252 = x;
        double r573253 = y;
        double r573254 = 4.0;
        double r573255 = r573253 * r573254;
        double r573256 = t;
        double r573257 = r573255 * r573256;
        double r573258 = z;
        double r573259 = -r573258;
        double r573260 = r573255 * r573259;
        double r573261 = r573260 * r573258;
        double r573262 = r573257 + r573261;
        double r573263 = 0.0;
        double r573264 = 1.5;
        double r573265 = pow(r573263, r573264);
        double r573266 = r573265 * r573265;
        double r573267 = cbrt(r573266);
        double r573268 = r573255 * r573267;
        double r573269 = r573262 + r573268;
        double r573270 = fma(r573252, r573252, r573269);
        return r573270;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

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

Derivation

  1. Initial program 6.0

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

  2. Simplified6.0

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

  4. Applied add-sqr-sqrt35.1

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

  5. Applied prod-diff35.1

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

  6. Applied distribute-lft-in35.1

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

  7. Simplified6.0

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

  9. Applied sub-neg6.0

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

  10. Applied distribute-lft-in6.0

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

  12. Applied distribute-lft-neg-in6.0

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

  13. Applied associate-*r*6.0

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

  15. Applied add-cbrt-cube15.3

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(-z, z, z \cdot z\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)\right) \cdot \mathsf{fma}\left(-z, z, z \cdot z\right)}}\right)\]

  16. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \sqrt[3]{\color{blue}{{0}^{\frac{3}{2}} \cdot {0}^{\frac{3}{2}}}}\right)\]

  17. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, x, \left(\left(y \cdot 4\right) \cdot t + \left(\left(y \cdot 4\right) \cdot \left(-z\right)\right) \cdot z\right) + \left(y \cdot 4\right) \cdot \sqrt[3]{{0}^{\frac{3}{2}} \cdot {0}^{\frac{3}{2}}}\right)\]

Reproduce

herbie shell --seed 2020100 +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))))