Average Error: 16.0 → 8.7
Time: 23.6s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.0569496526028927 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 2.83517124515135 \cdot 10^{+130}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.0569496526028927 \cdot 10^{+52}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 2.83517124515135 \cdot 10^{+130}:\\
\;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r26282502 = x;
        double r26282503 = y;
        double r26282504 = r26282502 + r26282503;
        double r26282505 = z;
        double r26282506 = t;
        double r26282507 = r26282505 - r26282506;
        double r26282508 = r26282507 * r26282503;
        double r26282509 = a;
        double r26282510 = r26282509 - r26282506;
        double r26282511 = r26282508 / r26282510;
        double r26282512 = r26282504 - r26282511;
        return r26282512;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r26282513 = t;
        double r26282514 = -1.0569496526028927e+52;
        bool r26282515 = r26282513 <= r26282514;
        double r26282516 = z;
        double r26282517 = r26282516 / r26282513;
        double r26282518 = y;
        double r26282519 = x;
        double r26282520 = fma(r26282517, r26282518, r26282519);
        double r26282521 = 2.83517124515135e+130;
        bool r26282522 = r26282513 <= r26282521;
        double r26282523 = r26282513 - r26282516;
        double r26282524 = a;
        double r26282525 = r26282524 - r26282513;
        double r26282526 = r26282518 / r26282525;
        double r26282527 = r26282523 * r26282526;
        double r26282528 = r26282518 + r26282519;
        double r26282529 = r26282527 + r26282528;
        double r26282530 = r26282522 ? r26282529 : r26282520;
        double r26282531 = r26282515 ? r26282520 : r26282530;
        return r26282531;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.0
Target8.2
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.0569496526028927e+52 or 2.83517124515135e+130 < t

    1. Initial program 28.8

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

    2. Simplified20.8

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

    3. Taylor expanded around inf 17.7

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

    4. Simplified12.8

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

    if -1.0569496526028927e+52 < t < 2.83517124515135e+130

    1. Initial program 8.2

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

    2. Simplified6.2

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

    4. Applied fma-udef6.2

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0569496526028927 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 2.83517124515135 \cdot 10^{+130}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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