Average Error: 16.7 → 8.5
Time: 23.1s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.384317655011651968640328382399140234507 \cdot 10^{125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 1.033502621996504461309739448189431813585 \cdot 10^{135}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{t - z}}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.384317655011651968640328382399140234507 \cdot 10^{125}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 1.033502621996504461309739448189431813585 \cdot 10^{135}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{t - z}}, 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 r24421766 = x;
        double r24421767 = y;
        double r24421768 = r24421766 + r24421767;
        double r24421769 = z;
        double r24421770 = t;
        double r24421771 = r24421769 - r24421770;
        double r24421772 = r24421771 * r24421767;
        double r24421773 = a;
        double r24421774 = r24421773 - r24421770;
        double r24421775 = r24421772 / r24421774;
        double r24421776 = r24421768 - r24421775;
        return r24421776;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24421777 = t;
        double r24421778 = -1.384317655011652e+125;
        bool r24421779 = r24421777 <= r24421778;
        double r24421780 = z;
        double r24421781 = r24421780 / r24421777;
        double r24421782 = y;
        double r24421783 = x;
        double r24421784 = fma(r24421781, r24421782, r24421783);
        double r24421785 = 1.0335026219965045e+135;
        bool r24421786 = r24421777 <= r24421785;
        double r24421787 = 1.0;
        double r24421788 = a;
        double r24421789 = r24421788 - r24421777;
        double r24421790 = r24421777 - r24421780;
        double r24421791 = r24421789 / r24421790;
        double r24421792 = r24421787 / r24421791;
        double r24421793 = r24421782 + r24421783;
        double r24421794 = fma(r24421782, r24421792, r24421793);
        double r24421795 = r24421786 ? r24421794 : r24421784;
        double r24421796 = r24421779 ? r24421784 : r24421795;
        return r24421796;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.7
Target8.6
Herbie8.5
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.384317655011652e+125 or 1.0335026219965045e+135 < t

    1. Initial program 31.6

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

    2. Simplified22.2

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

    3. Taylor expanded around inf 17.0

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

    4. Simplified11.6

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

    if -1.384317655011652e+125 < t < 1.0335026219965045e+135

    1. Initial program 9.8

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

    2. Simplified7.1

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

    4. Applied clear-num7.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{a - t}{t - z}}}, y + x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.384317655011651968640328382399140234507 \cdot 10^{125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 1.033502621996504461309739448189431813585 \cdot 10^{135}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{a - t}{t - z}}, y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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