Average Error: 15.7 → 8.1
Time: 12.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.157437085390881 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 1.0771781367201906 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.157437085390881 \cdot 10^{+172}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 1.0771781367201906 \cdot 10^{+108}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, 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 r9967496 = x;
        double r9967497 = y;
        double r9967498 = r9967496 + r9967497;
        double r9967499 = z;
        double r9967500 = t;
        double r9967501 = r9967499 - r9967500;
        double r9967502 = r9967501 * r9967497;
        double r9967503 = a;
        double r9967504 = r9967503 - r9967500;
        double r9967505 = r9967502 / r9967504;
        double r9967506 = r9967498 - r9967505;
        return r9967506;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r9967507 = t;
        double r9967508 = -6.157437085390881e+172;
        bool r9967509 = r9967507 <= r9967508;
        double r9967510 = z;
        double r9967511 = r9967510 / r9967507;
        double r9967512 = y;
        double r9967513 = x;
        double r9967514 = fma(r9967511, r9967512, r9967513);
        double r9967515 = 1.0771781367201906e+108;
        bool r9967516 = r9967507 <= r9967515;
        double r9967517 = r9967507 - r9967510;
        double r9967518 = a;
        double r9967519 = r9967518 - r9967507;
        double r9967520 = r9967517 / r9967519;
        double r9967521 = r9967512 + r9967513;
        double r9967522 = fma(r9967512, r9967520, r9967521);
        double r9967523 = r9967516 ? r9967522 : r9967514;
        double r9967524 = r9967509 ? r9967514 : r9967523;
        return r9967524;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original15.7
Target7.9
Herbie8.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -6.157437085390881e+172 or 1.0771781367201906e+108 < t

    1. Initial program 30.7

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

    2. Simplified21.0

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

    3. Taylor expanded around inf 16.3

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

    4. Simplified10.9

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

    if -6.157437085390881e+172 < t < 1.0771781367201906e+108

    1. Initial program 9.7

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

    2. Simplified7.0

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.157437085390881 \cdot 10^{+172}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 1.0771781367201906 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a - t}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))))