Average Error: 10.5 → 0.3
Time: 8.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -5.2646213116575603 \cdot 10^{290}:\\ \;\;\;\;y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right) + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.1606107956763068 \cdot 10^{295}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{z - a}{y}}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -5.2646213116575603 \cdot 10^{290}:\\
\;\;\;\;y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right) + x\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.1606107956763068 \cdot 10^{295}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{z - a}{y}}, z - t, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r729975 = x;
        double r729976 = y;
        double r729977 = z;
        double r729978 = t;
        double r729979 = r729977 - r729978;
        double r729980 = r729976 * r729979;
        double r729981 = a;
        double r729982 = r729977 - r729981;
        double r729983 = r729980 / r729982;
        double r729984 = r729975 + r729983;
        return r729984;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r729985 = y;
        double r729986 = z;
        double r729987 = t;
        double r729988 = r729986 - r729987;
        double r729989 = r729985 * r729988;
        double r729990 = a;
        double r729991 = r729986 - r729990;
        double r729992 = r729989 / r729991;
        double r729993 = -5.26462131165756e+290;
        bool r729994 = r729992 <= r729993;
        double r729995 = 1.0;
        double r729996 = r729995 / r729991;
        double r729997 = r729988 * r729996;
        double r729998 = r729985 * r729997;
        double r729999 = x;
        double r730000 = r729998 + r729999;
        double r730001 = 1.1606107956763068e+295;
        bool r730002 = r729992 <= r730001;
        double r730003 = r729992 + r729999;
        double r730004 = r729991 / r729985;
        double r730005 = r729995 / r730004;
        double r730006 = fma(r730005, r729988, r729999);
        double r730007 = r730002 ? r730003 : r730006;
        double r730008 = r729994 ? r730000 : r730007;
        return r730008;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original10.5
Target1.3
Herbie0.3
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) (- z a)) < -5.26462131165756e+290

    1. Initial program 61.2

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

    2. Simplified1.0

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

    4. Applied fma-udef1.0

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

    6. Applied div-inv1.2

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

    7. Applied associate-*l*0.7

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

    8. Simplified0.6

      \[\leadsto y \cdot \color{blue}{\frac{z - t}{z - a}} + x\]
    9. Using strategy rm

    10. Applied div-inv0.7

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

    if -5.26462131165756e+290 < (/ (* y (- z t)) (- z a)) < 1.1606107956763068e+295

    1. Initial program 0.2

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

    2. Simplified3.3

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

    4. Applied fma-udef3.3

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

    6. Applied associate-*l/0.2

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

    if 1.1606107956763068e+295 < (/ (* y (- z t)) (- z a))

    1. Initial program 62.1

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

    2. Simplified0.6

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

    4. Applied clear-num0.7

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -5.2646213116575603 \cdot 10^{290}:\\ \;\;\;\;y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right) + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.1606107956763068 \cdot 10^{295}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{z - a}{y}}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))