Average Error: 2.0 → 2.4
Time: 5.2s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.370918042146381127285332195867531032316 \cdot 10^{-76} \lor \neg \left(x \le 1.78843006131791689904510956091429861366 \cdot 10^{49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.370918042146381127285332195867531032316 \cdot 10^{-76} \lor \neg \left(x \le 1.78843006131791689904510956091429861366 \cdot 10^{49}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + 1 \cdot \frac{\left(y - x\right) \cdot z}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r668979 = x;
        double r668980 = y;
        double r668981 = r668980 - r668979;
        double r668982 = z;
        double r668983 = t;
        double r668984 = r668982 / r668983;
        double r668985 = r668981 * r668984;
        double r668986 = r668979 + r668985;
        return r668986;
}
↓
double f(double x, double y, double z, double t) {
        double r668987 = x;
        double r668988 = -1.3709180421463811e-76;
        bool r668989 = r668987 <= r668988;
        double r668990 = 1.788430061317917e+49;
        bool r668991 = r668987 <= r668990;
        double r668992 = !r668991;
        bool r668993 = r668989 || r668992;
        double r668994 = z;
        double r668995 = t;
        double r668996 = r668994 / r668995;
        double r668997 = y;
        double r668998 = r668997 - r668987;
        double r668999 = fma(r668996, r668998, r668987);
        double r669000 = 1.0;
        double r669001 = r668998 * r668994;
        double r669002 = r669001 / r668995;
        double r669003 = r669000 * r669002;
        double r669004 = r668987 + r669003;
        double r669005 = r668993 ? r668999 : r669004;
        return r669005;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.0
Target2.2
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.88671875:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3709180421463811e-76 or 1.788430061317917e+49 < x

    1. Initial program 0.2

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

    3. Applied *-commutative0.2

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

    4. Taylor expanded around 0 9.2

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

    5. Simplified0.2

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

    if -1.3709180421463811e-76 < x < 1.788430061317917e+49

    1. Initial program 3.6

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

    3. Applied *-commutative3.6

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

    5. Applied *-un-lft-identity3.6

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

    6. Applied associate-*l*3.6

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

    7. Simplified4.4

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.370918042146381127285332195867531032316 \cdot 10^{-76} \lor \neg \left(x \le 1.78843006131791689904510956091429861366 \cdot 10^{49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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