Average Error: 16.2 → 10.7
Time: 3.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.24704184831236509 \cdot 10^{182}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 6.6578301054235618 \cdot 10^{155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.24704184831236509 \cdot 10^{182}:\\
\;\;\;\;x\\

\mathbf{elif}\;t \le 6.6578301054235618 \cdot 10^{155}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r524606 = x;
        double r524607 = y;
        double r524608 = r524606 + r524607;
        double r524609 = z;
        double r524610 = t;
        double r524611 = r524609 - r524610;
        double r524612 = r524611 * r524607;
        double r524613 = a;
        double r524614 = r524613 - r524610;
        double r524615 = r524612 / r524614;
        double r524616 = r524608 - r524615;
        return r524616;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r524617 = t;
        double r524618 = -1.2470418483123651e+182;
        bool r524619 = r524617 <= r524618;
        double r524620 = x;
        double r524621 = 6.657830105423562e+155;
        bool r524622 = r524617 <= r524621;
        double r524623 = y;
        double r524624 = a;
        double r524625 = r524624 - r524617;
        double r524626 = r524623 / r524625;
        double r524627 = z;
        double r524628 = r524617 - r524627;
        double r524629 = r524620 + r524623;
        double r524630 = fma(r524626, r524628, r524629);
        double r524631 = r524622 ? r524630 : r524620;
        double r524632 = r524619 ? r524620 : r524631;
        return r524632;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.2
Target8.5
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.2470418483123651e+182 or 6.657830105423562e+155 < t

    1. Initial program 33.4

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

    2. Simplified24.9

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

    3. Taylor expanded around 0 20.3

      \[\leadsto \color{blue}{x}\]

    if -1.2470418483123651e+182 < t < 6.657830105423562e+155

    1. Initial program 10.7

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

    2. Simplified7.6

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

    4. Applied div-inv7.7

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

    6. Applied un-div-inv7.6

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.24704184831236509 \cdot 10^{182}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \le 6.6578301054235618 \cdot 10^{155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :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))))