Average Error: 1.9 → 2.0
Time: 15.3s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\mathsf{fma}\left(y, z, \mathsf{fma}\left(t, a, x\right)\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\right) \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\mathsf{fma}\left(y, z, \mathsf{fma}\left(t, a, x\right)\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\right) \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r27952174 = x;
        double r27952175 = y;
        double r27952176 = z;
        double r27952177 = r27952175 * r27952176;
        double r27952178 = r27952174 + r27952177;
        double r27952179 = t;
        double r27952180 = a;
        double r27952181 = r27952179 * r27952180;
        double r27952182 = r27952178 + r27952181;
        double r27952183 = r27952180 * r27952176;
        double r27952184 = b;
        double r27952185 = r27952183 * r27952184;
        double r27952186 = r27952182 + r27952185;
        return r27952186;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r27952187 = y;
        double r27952188 = z;
        double r27952189 = t;
        double r27952190 = a;
        double r27952191 = x;
        double r27952192 = fma(r27952189, r27952190, r27952191);
        double r27952193 = fma(r27952187, r27952188, r27952192);
        double r27952194 = r27952190 * r27952188;
        double r27952195 = b;
        double r27952196 = r27952194 * r27952195;
        double r27952197 = cbrt(r27952196);
        double r27952198 = r27952197 * r27952197;
        double r27952199 = cbrt(r27952195);
        double r27952200 = r27952199 * r27952199;
        double r27952201 = r27952194 * r27952200;
        double r27952202 = r27952199 * r27952201;
        double r27952203 = cbrt(r27952202);
        double r27952204 = r27952198 * r27952203;
        double r27952205 = r27952193 + r27952204;
        return r27952205;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original1.9
Target0.3
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888128:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.758974318836428710669076838657752600596 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Initial program 1.9

    \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

  2. Taylor expanded around inf 1.9

    \[\leadsto \color{blue}{\left(x + \left(z \cdot y + a \cdot t\right)\right)} + \left(a \cdot z\right) \cdot b\]

  3. Simplified1.9

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

  5. Applied add-cube-cbrt2.0

    \[\leadsto \mathsf{fma}\left(y, z, \mathsf{fma}\left(t, a, x\right)\right) + \color{blue}{\left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\right) \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt2.0

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

  8. Applied associate-*r*2.0

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

  9. Final simplification2.0

    \[\leadsto \mathsf{fma}\left(y, z, \mathsf{fma}\left(t, a, x\right)\right) + \left(\sqrt[3]{\left(a \cdot z\right) \cdot b} \cdot \sqrt[3]{\left(a \cdot z\right) \cdot b}\right) \cdot \sqrt[3]{\sqrt[3]{b} \cdot \left(\left(a \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"

  :herbie-target
  (if (< z -1.1820553527347888e+19) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))