Average Error: 10.9 → 1.0
Time: 26.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.73706509850903537 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \le 5.1161288039363663 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.73706509850903537 \cdot 10^{-97}:\\
\;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\

\mathbf{elif}\;y \le 5.1161288039363663 \cdot 10^{-19}:\\
\;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r444943 = x;
        double r444944 = y;
        double r444945 = z;
        double r444946 = t;
        double r444947 = r444945 - r444946;
        double r444948 = r444944 * r444947;
        double r444949 = a;
        double r444950 = r444945 - r444949;
        double r444951 = r444948 / r444950;
        double r444952 = r444943 + r444951;
        return r444952;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r444953 = y;
        double r444954 = -5.737065098509035e-97;
        bool r444955 = r444953 <= r444954;
        double r444956 = x;
        double r444957 = z;
        double r444958 = a;
        double r444959 = r444957 - r444958;
        double r444960 = t;
        double r444961 = r444957 - r444960;
        double r444962 = r444959 / r444961;
        double r444963 = r444953 / r444962;
        double r444964 = r444956 + r444963;
        double r444965 = 5.116128803936366e-19;
        bool r444966 = r444953 <= r444965;
        double r444967 = 1.0;
        double r444968 = r444967 / r444959;
        double r444969 = r444953 * r444961;
        double r444970 = r444968 * r444969;
        double r444971 = r444956 + r444970;
        double r444972 = r444953 / r444959;
        double r444973 = fma(r444972, r444961, r444956);
        double r444974 = r444966 ? r444971 : r444973;
        double r444975 = r444955 ? r444964 : r444974;
        return r444975;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.2
Herbie1.0
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -5.737065098509035e-97

    1. Initial program 17.7

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

    3. Applied associate-/l*0.6

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

    if -5.737065098509035e-97 < y < 5.116128803936366e-19

    1. Initial program 0.2

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

    3. Applied associate-/l*2.0

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

    5. Applied clear-num2.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{z - a}{z - t}}{y}}}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity2.0

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

    8. Applied div-inv2.1

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

    9. Applied times-frac0.4

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

    10. Applied add-cube-cbrt0.4

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

    11. Applied times-frac0.3

      \[\leadsto x + \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{z - a}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{1}{z - t}}{y}}}\]

    12. Simplified0.3

      \[\leadsto x + \color{blue}{\frac{1}{z - a}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{1}{z - t}}{y}}\]

    13. Simplified0.2

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

    if 5.116128803936366e-19 < y

    1. Initial program 21.5

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

    2. Simplified3.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.73706509850903537 \cdot 10^{-97}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \le 5.1161288039363663 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{z - a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"

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

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