Average Error: 16.3 → 9.1
Time: 28.0s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.36754662391784781499062445922864415335 \cdot 10^{55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 9.944888148073560836094682457240141243033 \cdot 10^{179}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -4.36754662391784781499062445922864415335 \cdot 10^{55}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 9.944888148073560836094682457240141243033 \cdot 10^{179}:\\
\;\;\;\;\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 r23568979 = x;
        double r23568980 = y;
        double r23568981 = r23568979 + r23568980;
        double r23568982 = z;
        double r23568983 = t;
        double r23568984 = r23568982 - r23568983;
        double r23568985 = r23568984 * r23568980;
        double r23568986 = a;
        double r23568987 = r23568986 - r23568983;
        double r23568988 = r23568985 / r23568987;
        double r23568989 = r23568981 - r23568988;
        return r23568989;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23568990 = t;
        double r23568991 = -4.367546623917848e+55;
        bool r23568992 = r23568990 <= r23568991;
        double r23568993 = z;
        double r23568994 = r23568993 / r23568990;
        double r23568995 = y;
        double r23568996 = x;
        double r23568997 = fma(r23568994, r23568995, r23568996);
        double r23568998 = 9.944888148073561e+179;
        bool r23568999 = r23568990 <= r23568998;
        double r23569000 = r23568990 - r23568993;
        double r23569001 = a;
        double r23569002 = r23569001 - r23568990;
        double r23569003 = r23568995 / r23569002;
        double r23569004 = r23569000 * r23569003;
        double r23569005 = r23568995 + r23568996;
        double r23569006 = r23569004 + r23569005;
        double r23569007 = r23568999 ? r23569006 : r23568997;
        double r23569008 = r23568992 ? r23568997 : r23569007;
        return r23569008;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.3
Target8.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.367546623917848e+55 or 9.944888148073561e+179 < t

    1. Initial program 29.9

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

    2. Simplified21.9

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

    3. Taylor expanded around inf 17.8

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

    4. Simplified12.7

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

    if -4.367546623917848e+55 < t < 9.944888148073561e+179

    1. Initial program 9.7

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

    2. Simplified7.3

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

    4. Applied fma-udef7.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.36754662391784781499062445922864415335 \cdot 10^{55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 9.944888148073560836094682457240141243033 \cdot 10^{179}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))