Average Error: 3.7 → 0.8
Time: 13.3s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -1.284011004020568602828927919392572216693 \cdot 10^{73}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 3.230635861036458324525646415982268595864 \cdot 10^{-76}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot t\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;y \cdot 9 \le -1.284011004020568602828927919392572216693 \cdot 10^{73}:\\
\;\;\;\;\mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\right)\\

\mathbf{elif}\;y \cdot 9 \le 3.230635861036458324525646415982268595864 \cdot 10^{-76}:\\
\;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot t\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r487179 = x;
        double r487180 = 2.0;
        double r487181 = r487179 * r487180;
        double r487182 = y;
        double r487183 = 9.0;
        double r487184 = r487182 * r487183;
        double r487185 = z;
        double r487186 = r487184 * r487185;
        double r487187 = t;
        double r487188 = r487186 * r487187;
        double r487189 = r487181 - r487188;
        double r487190 = a;
        double r487191 = 27.0;
        double r487192 = r487190 * r487191;
        double r487193 = b;
        double r487194 = r487192 * r487193;
        double r487195 = r487189 + r487194;
        return r487195;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r487196 = y;
        double r487197 = 9.0;
        double r487198 = r487196 * r487197;
        double r487199 = -1.2840110040205686e+73;
        bool r487200 = r487198 <= r487199;
        double r487201 = x;
        double r487202 = 2.0;
        double r487203 = t;
        double r487204 = z;
        double r487205 = r487203 * r487204;
        double r487206 = r487205 * r487198;
        double r487207 = -r487206;
        double r487208 = fma(r487201, r487202, r487207);
        double r487209 = -r487196;
        double r487210 = fma(r487197, r487209, r487198);
        double r487211 = 27.0;
        double r487212 = sqrt(r487211);
        double r487213 = a;
        double r487214 = b;
        double r487215 = r487213 * r487214;
        double r487216 = r487212 * r487215;
        double r487217 = r487212 * r487216;
        double r487218 = fma(r487205, r487210, r487217);
        double r487219 = r487208 + r487218;
        double r487220 = 3.2306358610364583e-76;
        bool r487221 = r487198 <= r487220;
        double r487222 = r487201 * r487202;
        double r487223 = r487198 * r487203;
        double r487224 = r487223 * r487204;
        double r487225 = r487222 - r487224;
        double r487226 = r487213 * r487211;
        double r487227 = r487226 * r487214;
        double r487228 = r487225 + r487227;
        double r487229 = r487197 * r487205;
        double r487230 = r487196 * r487229;
        double r487231 = r487222 - r487230;
        double r487232 = r487231 + r487227;
        double r487233 = r487221 ? r487228 : r487232;
        double r487234 = r487200 ? r487219 : r487233;
        return r487234;
}

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

Original3.7
Target2.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y 9.0) < -1.2840110040205686e+73

    1. Initial program 9.9

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*1.2

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified1.2

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]
    5. Using strategy rm

    6. Applied prod-diff1.2

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(-t \cdot z, y \cdot 9, \left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right)\right)} + \left(a \cdot 27\right) \cdot b\]

    7. Applied associate-+l+1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \left(\mathsf{fma}\left(-t \cdot z, y \cdot 9, \left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \left(a \cdot 27\right) \cdot b\right)}\]

    8. Simplified1.1

      \[\leadsto \mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \color{blue}{\mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), 27 \cdot \left(a \cdot b\right)\right)}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt1.1

      \[\leadsto \mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), \color{blue}{\left(\sqrt{27} \cdot \sqrt{27}\right)} \cdot \left(a \cdot b\right)\right)\]

    11. Applied associate-*l*1.2

      \[\leadsto \mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), \color{blue}{\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)}\right)\]

    if -1.2840110040205686e+73 < (* y 9.0) < 3.2306358610364583e-76

    1. Initial program 0.8

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*5.5

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified5.5

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]
    5. Using strategy rm

    6. Applied associate-*r*0.6

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(y \cdot 9\right) \cdot t\right) \cdot z}\right) + \left(a \cdot 27\right) \cdot b\]

    if 3.2306358610364583e-76 < (* y 9.0)

    1. Initial program 6.2

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*1.1

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified1.1

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]
    5. Using strategy rm

    6. Applied associate-*l*1.0

      \[\leadsto \left(x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(t \cdot z\right)\right)}\right) + \left(a \cdot 27\right) \cdot b\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -1.284011004020568602828927919392572216693 \cdot 10^{73}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, -\left(t \cdot z\right) \cdot \left(y \cdot 9\right)\right) + \mathsf{fma}\left(t \cdot z, \mathsf{fma}\left(9, -y, y \cdot 9\right), \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 3.230635861036458324525646415982268595864 \cdot 10^{-76}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot t\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))