Average Error: 3.7 → 1.6
Time: 11.5s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{t}{z}}{y \cdot 3}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{t}{z}}{y \cdot 3}
double f(double x, double y, double z, double t) {
        double r965145 = x;
        double r965146 = y;
        double r965147 = z;
        double r965148 = 3.0;
        double r965149 = r965147 * r965148;
        double r965150 = r965146 / r965149;
        double r965151 = r965145 - r965150;
        double r965152 = t;
        double r965153 = r965149 * r965146;
        double r965154 = r965152 / r965153;
        double r965155 = r965151 + r965154;
        return r965155;
}
↓
double f(double x, double y, double z, double t) {
        double r965156 = 1.0;
        double r965157 = x;
        double r965158 = y;
        double r965159 = 3.0;
        double r965160 = r965158 / r965159;
        double r965161 = -r965160;
        double r965162 = z;
        double r965163 = r965161 / r965162;
        double r965164 = fma(r965156, r965157, r965163);
        double r965165 = t;
        double r965166 = r965165 / r965162;
        double r965167 = r965158 * r965159;
        double r965168 = r965166 / r965167;
        double r965169 = r965164 + r965168;
        return r965169;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.7
Target1.6
Herbie1.6
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Initial program 3.7

    \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
  2. Using strategy rm

  3. Applied associate-/r*1.6

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

  5. Applied associate-/r*1.6

    \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{\frac{t}{z}}{3}}}{y}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity1.6

    \[\leadsto \left(\color{blue}{1 \cdot x} - \frac{y}{z \cdot 3}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  8. Applied fma-neg1.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\frac{y}{z \cdot 3}\right)} + \frac{\frac{\frac{t}{z}}{3}}{y}\]

  9. Simplified1.6

    \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{-\frac{y}{3}}{z}}\right) + \frac{\frac{\frac{t}{z}}{3}}{y}\]
  10. Using strategy rm

  11. Applied div-inv1.6

    \[\leadsto \mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\color{blue}{\frac{t}{z} \cdot \frac{1}{3}}}{y}\]

  12. Applied associate-/l*1.6

    \[\leadsto \mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \color{blue}{\frac{\frac{t}{z}}{\frac{y}{\frac{1}{3}}}}\]

  13. Simplified1.6

    \[\leadsto \mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{t}{z}}{\color{blue}{y \cdot 3}}\]

  14. Final simplification1.6

    \[\leadsto \mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{t}{z}}{y \cdot 3}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))