Average Error: 1.7 → 0.6
Time: 18.7s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -46722405.0959222018718719482421875:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \mathbf{elif}\;y \le 1.976628579682119090515935104236535804433 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -46722405.0959222018718719482421875:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\

\mathbf{elif}\;y \le 1.976628579682119090515935104236535804433 \cdot 10^{-101}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r502786 = x;
        double r502787 = y;
        double r502788 = z;
        double r502789 = t;
        double r502790 = r502788 - r502789;
        double r502791 = a;
        double r502792 = r502788 - r502791;
        double r502793 = r502790 / r502792;
        double r502794 = r502787 * r502793;
        double r502795 = r502786 + r502794;
        return r502795;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r502796 = y;
        double r502797 = -46722405.0959222;
        bool r502798 = r502796 <= r502797;
        double r502799 = z;
        double r502800 = t;
        double r502801 = r502799 - r502800;
        double r502802 = a;
        double r502803 = r502799 - r502802;
        double r502804 = r502801 / r502803;
        double r502805 = x;
        double r502806 = fma(r502804, r502796, r502805);
        double r502807 = 1.976628579682119e-101;
        bool r502808 = r502796 <= r502807;
        double r502809 = r502801 * r502796;
        double r502810 = r502809 / r502803;
        double r502811 = r502810 + r502805;
        double r502812 = 1.0;
        double r502813 = r502812 / r502803;
        double r502814 = r502801 * r502813;
        double r502815 = fma(r502814, r502796, r502805);
        double r502816 = r502808 ? r502811 : r502815;
        double r502817 = r502798 ? r502806 : r502816;
        return r502817;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.7
Target1.5
Herbie0.6
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -46722405.0959222

    1. Initial program 0.8

      \[x + y \cdot \frac{z - t}{z - a}\]

    2. Simplified0.8

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

    if -46722405.0959222 < y < 1.976628579682119e-101

    1. Initial program 2.7

      \[x + y \cdot \frac{z - t}{z - a}\]

    2. Simplified2.7

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

    4. Applied div-inv2.8

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

    6. Applied add-cube-cbrt3.0

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

    7. Applied associate-/r*3.0

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

    9. Applied fma-udef3.0

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

    10. Simplified0.4

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

    if 1.976628579682119e-101 < y

    1. Initial program 0.7

      \[x + y \cdot \frac{z - t}{z - a}\]

    2. Simplified0.7

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

    4. Applied div-inv0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -46722405.0959222018718719482421875:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z - a}, y, x\right)\\ \mathbf{elif}\;y \le 1.976628579682119090515935104236535804433 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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