Average Error: 10.4 → 0.2
Time: 20.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y}} - \left(\frac{t}{\frac{a - t}{y}} - x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.105950757333242047287578696854372670873 \cdot 10^{306}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} = -\infty:\\
\;\;\;\;\frac{z}{\frac{a - t}{y}} - \left(\frac{t}{\frac{a - t}{y}} - x\right)\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.105950757333242047287578696854372670873 \cdot 10^{306}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r333922 = x;
        double r333923 = y;
        double r333924 = z;
        double r333925 = t;
        double r333926 = r333924 - r333925;
        double r333927 = r333923 * r333926;
        double r333928 = a;
        double r333929 = r333928 - r333925;
        double r333930 = r333927 / r333929;
        double r333931 = r333922 + r333930;
        return r333931;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r333932 = y;
        double r333933 = z;
        double r333934 = t;
        double r333935 = r333933 - r333934;
        double r333936 = r333932 * r333935;
        double r333937 = a;
        double r333938 = r333937 - r333934;
        double r333939 = r333936 / r333938;
        double r333940 = -inf.0;
        bool r333941 = r333939 <= r333940;
        double r333942 = r333938 / r333932;
        double r333943 = r333933 / r333942;
        double r333944 = r333934 / r333942;
        double r333945 = x;
        double r333946 = r333944 - r333945;
        double r333947 = r333943 - r333946;
        double r333948 = 5.105950757333242e+306;
        bool r333949 = r333939 <= r333948;
        double r333950 = r333945 + r333939;
        double r333951 = r333932 / r333938;
        double r333952 = fma(r333951, r333935, r333945);
        double r333953 = r333949 ? r333950 : r333952;
        double r333954 = r333941 ? r333947 : r333953;
        return r333954;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target1.5
Herbie0.2
\[x + \frac{y}{\frac{a - t}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- a t)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

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

    4. Applied clear-num0.3

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

    6. Applied fma-udef0.3

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

    7. Simplified0.2

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a - t}{y}}} + x\]
    8. Using strategy rm

    9. Applied div-sub0.2

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

    10. Applied associate-+l-0.2

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

    if -inf.0 < (/ (* y (- z t)) (- a t)) < 5.105950757333242e+306

    1. Initial program 0.2

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

    if 5.105950757333242e+306 < (/ (* y (- z t)) (- a t))

    1. Initial program 63.6

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

    2. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y}} - \left(\frac{t}{\frac{a - t}{y}} - x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a - t} \le 5.105950757333242047287578696854372670873 \cdot 10^{306}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

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

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