Average Error: 3.6 → 1.9
Time: 13.4s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4407887380689837301114156875776:\\ \;\;\;\;\left(x - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4407887380689837301114156875776:\\
\;\;\;\;\left(x - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r823665 = x;
        double r823666 = y;
        double r823667 = z;
        double r823668 = 3.0;
        double r823669 = r823667 * r823668;
        double r823670 = r823666 / r823669;
        double r823671 = r823665 - r823670;
        double r823672 = t;
        double r823673 = r823669 * r823666;
        double r823674 = r823672 / r823673;
        double r823675 = r823671 + r823674;
        return r823675;
}
↓
double f(double x, double y, double z, double t) {
        double r823676 = t;
        double r823677 = -4.4078873806898373e+30;
        bool r823678 = r823676 <= r823677;
        double r823679 = x;
        double r823680 = 0.3333333333333333;
        double r823681 = y;
        double r823682 = z;
        double r823683 = r823681 / r823682;
        double r823684 = r823680 * r823683;
        double r823685 = r823679 - r823684;
        double r823686 = 3.0;
        double r823687 = r823682 * r823686;
        double r823688 = r823687 * r823681;
        double r823689 = r823676 / r823688;
        double r823690 = r823685 + r823689;
        double r823691 = 1.0;
        double r823692 = r823681 / r823686;
        double r823693 = -r823692;
        double r823694 = r823693 / r823682;
        double r823695 = fma(r823691, r823679, r823694);
        double r823696 = r823691 / r823682;
        double r823697 = r823676 / r823686;
        double r823698 = r823681 / r823697;
        double r823699 = r823696 / r823698;
        double r823700 = r823695 + r823699;
        double r823701 = r823678 ? r823690 : r823700;
        return r823701;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original3.6
Target1.7
Herbie1.9
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.4078873806898373e+30

    1. Initial program 0.6

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

    2. Taylor expanded around 0 0.7

      \[\leadsto \left(x - \color{blue}{0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]

    if -4.4078873806898373e+30 < t

    1. Initial program 4.2

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

    3. Applied associate-/r*1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied times-frac1.5

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

    7. Applied associate-/l*2.3

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

    9. Applied *-un-lft-identity2.3

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

    10. Applied fma-neg2.3

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

    11. Simplified2.2

      \[\leadsto \mathsf{fma}\left(1, x, \color{blue}{\frac{-\frac{y}{3}}{z}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4407887380689837301114156875776:\\ \;\;\;\;\left(x - 0.3333333333333333148296162562473909929395 \cdot \frac{y}{z}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, \frac{-\frac{y}{3}}{z}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))))