Average Error: 16.3 → 8.6
Time: 21.2s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.900002769274410974322265838870386622475 \cdot 10^{-91} \lor \neg \left(a \le 2.657500584912369811224575444645349757225 \cdot 10^{-166}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{a - t} \cdot \left(t - z\right), y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.900002769274410974322265838870386622475 \cdot 10^{-91} \lor \neg \left(a \le 2.657500584912369811224575444645349757225 \cdot 10^{-166}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{a - t} \cdot \left(t - z\right), y + x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r473736 = x;
        double r473737 = y;
        double r473738 = r473736 + r473737;
        double r473739 = z;
        double r473740 = t;
        double r473741 = r473739 - r473740;
        double r473742 = r473741 * r473737;
        double r473743 = a;
        double r473744 = r473743 - r473740;
        double r473745 = r473742 / r473744;
        double r473746 = r473738 - r473745;
        return r473746;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r473747 = a;
        double r473748 = -2.900002769274411e-91;
        bool r473749 = r473747 <= r473748;
        double r473750 = 2.6575005849123698e-166;
        bool r473751 = r473747 <= r473750;
        double r473752 = !r473751;
        bool r473753 = r473749 || r473752;
        double r473754 = y;
        double r473755 = 1.0;
        double r473756 = t;
        double r473757 = r473747 - r473756;
        double r473758 = r473755 / r473757;
        double r473759 = z;
        double r473760 = r473756 - r473759;
        double r473761 = r473758 * r473760;
        double r473762 = x;
        double r473763 = r473754 + r473762;
        double r473764 = fma(r473754, r473761, r473763);
        double r473765 = r473759 / r473756;
        double r473766 = fma(r473754, r473765, r473762);
        double r473767 = r473753 ? r473764 : r473766;
        return r473767;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.3
Target8.6
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.900002769274411e-91 or 2.6575005849123698e-166 < a

    1. Initial program 14.8

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

    2. Simplified8.2

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

    4. Applied div-inv8.2

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

    if -2.900002769274411e-91 < a < 2.6575005849123698e-166

    1. Initial program 20.3

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

    2. Simplified19.2

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

    4. Applied div-inv19.2

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

    6. Applied add-cube-cbrt19.5

      \[\leadsto \mathsf{fma}\left(y, \left(t - z\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right) \cdot \sqrt[3]{\frac{1}{a - t}}\right)}, y + x\right)\]

    7. Taylor expanded around inf 10.7

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

    8. Simplified9.7

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.900002769274410974322265838870386622475 \cdot 10^{-91} \lor \neg \left(a \le 2.657500584912369811224575444645349757225 \cdot 10^{-166}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{a - t} \cdot \left(t - z\right), y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))))