Average Error: 1.9 → 2.0
Time: 15.5s
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 r25767195 = x;
        double r25767196 = y;
        double r25767197 = z;
        double r25767198 = r25767196 * r25767197;
        double r25767199 = r25767195 + r25767198;
        double r25767200 = t;
        double r25767201 = a;
        double r25767202 = r25767200 * r25767201;
        double r25767203 = r25767199 + r25767202;
        double r25767204 = r25767201 * r25767197;
        double r25767205 = b;
        double r25767206 = r25767204 * r25767205;
        double r25767207 = r25767203 + r25767206;
        return r25767207;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r25767208 = y;
        double r25767209 = z;
        double r25767210 = t;
        double r25767211 = a;
        double r25767212 = x;
        double r25767213 = fma(r25767210, r25767211, r25767212);
        double r25767214 = fma(r25767208, r25767209, r25767213);
        double r25767215 = r25767211 * r25767209;
        double r25767216 = b;
        double r25767217 = r25767215 * r25767216;
        double r25767218 = cbrt(r25767217);
        double r25767219 = r25767218 * r25767218;
        double r25767220 = cbrt(r25767216);
        double r25767221 = r25767220 * r25767220;
        double r25767222 = r25767215 * r25767221;
        double r25767223 = r25767220 * r25767222;
        double r25767224 = cbrt(r25767223);
        double r25767225 = r25767219 * r25767224;
        double r25767226 = r25767214 + r25767225;
        return r25767226;
}

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)))