Average Error: 10.5 → 0.3
Time: 10.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \le -5.2646213116575603 \cdot 10^{290}:\\ \;\;\;\;y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right) + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \le 1.1606107956763068 \cdot 10^{295}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{z - a}{y}}, z - t, x\right)\\ \end{array}\]
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}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\

\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 r559440 = x;
        double r559441 = y;
        double r559442 = z;
        double r559443 = t;
        double r559444 = r559442 - r559443;
        double r559445 = r559441 * r559444;
        double r559446 = a;
        double r559447 = r559442 - r559446;
        double r559448 = r559445 / r559447;
        double r559449 = r559440 + r559448;
        return r559449;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r559450 = y;
        double r559451 = z;
        double r559452 = t;
        double r559453 = r559451 - r559452;
        double r559454 = r559450 * r559453;
        double r559455 = a;
        double r559456 = r559451 - r559455;
        double r559457 = r559454 / r559456;
        double r559458 = -5.26462131165756e+290;
        bool r559459 = r559457 <= r559458;
        double r559460 = 1.0;
        double r559461 = r559460 / r559456;
        double r559462 = r559453 * r559461;
        double r559463 = r559450 * r559462;
        double r559464 = x;
        double r559465 = r559463 + r559464;
        double r559466 = 1.1606107956763068e+295;
        bool r559467 = r559457 <= r559466;
        double r559468 = r559464 + r559457;
        double r559469 = r559456 / r559450;
        double r559470 = r559460 / r559469;
        double r559471 = fma(r559470, r559453, r559464);
        double r559472 = r559467 ? r559468 : r559471;
        double r559473 = r559459 ? r559465 : r559472;
        return r559473;
}

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 pow13.3

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

    7. Applied pow13.3

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

    8. Applied pow-prod-down3.3

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

    9. Simplified0.2

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

    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

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

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))))