Average Error: 10.9 → 0.7
Time: 25.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.109016302886835245235186766998728474519 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \mathbf{elif}\;y \le 5.380855567614572390769079808720250862406 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.109016302886835245235186766998728474519 \cdot 10^{-147}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\

\mathbf{elif}\;y \le 5.380855567614572390769079808720250862406 \cdot 10^{-168}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r24468024 = x;
        double r24468025 = y;
        double r24468026 = z;
        double r24468027 = t;
        double r24468028 = r24468026 - r24468027;
        double r24468029 = r24468025 * r24468028;
        double r24468030 = a;
        double r24468031 = r24468026 - r24468030;
        double r24468032 = r24468029 / r24468031;
        double r24468033 = r24468024 + r24468032;
        return r24468033;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24468034 = y;
        double r24468035 = -1.1090163028868352e-147;
        bool r24468036 = r24468034 <= r24468035;
        double r24468037 = z;
        double r24468038 = t;
        double r24468039 = r24468037 - r24468038;
        double r24468040 = a;
        double r24468041 = r24468037 - r24468040;
        double r24468042 = r24468039 / r24468041;
        double r24468043 = x;
        double r24468044 = fma(r24468042, r24468034, r24468043);
        double r24468045 = 5.380855567614572e-168;
        bool r24468046 = r24468034 <= r24468045;
        double r24468047 = r24468039 * r24468034;
        double r24468048 = r24468047 / r24468041;
        double r24468049 = r24468048 + r24468043;
        double r24468050 = r24468046 ? r24468049 : r24468044;
        double r24468051 = r24468036 ? r24468044 : r24468050;
        return r24468051;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.9
Target1.1
Herbie0.7
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.1090163028868352e-147 or 5.380855567614572e-168 < y

    1. Initial program 14.7

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

    2. Simplified0.8

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

    if -1.1090163028868352e-147 < y < 5.380855567614572e-168

    1. Initial program 0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.109016302886835245235186766998728474519 \cdot 10^{-147}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \mathbf{elif}\;y \le 5.380855567614572390769079808720250862406 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +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))))