Average Error: 10.4 → 0.2
Time: 3.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.4622522651968869 \cdot 10^{301}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.4622522651968869 \cdot 10^{301}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z - t}{z - a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r558992 = x;
        double r558993 = y;
        double r558994 = z;
        double r558995 = t;
        double r558996 = r558994 - r558995;
        double r558997 = r558993 * r558996;
        double r558998 = a;
        double r558999 = r558994 - r558998;
        double r559000 = r558997 / r558999;
        double r559001 = r558992 + r559000;
        return r559001;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r559002 = y;
        double r559003 = z;
        double r559004 = t;
        double r559005 = r559003 - r559004;
        double r559006 = r559002 * r559005;
        double r559007 = a;
        double r559008 = r559003 - r559007;
        double r559009 = r559006 / r559008;
        double r559010 = -inf.0;
        bool r559011 = r559009 <= r559010;
        double r559012 = r559002 / r559008;
        double r559013 = x;
        double r559014 = fma(r559012, r559005, r559013);
        double r559015 = 1.462252265196887e+301;
        bool r559016 = r559009 <= r559015;
        double r559017 = r559013 + r559009;
        double r559018 = r559005 / r559008;
        double r559019 = r559002 * r559018;
        double r559020 = r559013 + r559019;
        double r559021 = r559016 ? r559017 : r559020;
        double r559022 = r559011 ? r559014 : r559021;
        return r559022;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.4
Target1.1
Herbie0.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- z a)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

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

    if -inf.0 < (/ (* y (- z t)) (- z a)) < 1.462252265196887e+301

    1. Initial program 0.2

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

    if 1.462252265196887e+301 < (/ (* y (- z t)) (- z a))

    1. Initial program 62.7

      \[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity62.7

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

    4. Applied times-frac0.2

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - t}{z - a}}\]

    5. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.4622522651968869 \cdot 10^{301}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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