Average Error: 1.3 → 0.5
Time: 23.2s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.776587407336161672094508528940042719675 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;y \le 2.143447732517704601694518276826181085268 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z - a}, \frac{y}{\frac{1}{z - t}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.776587407336161672094508528940042719675 \cdot 10^{-59}:\\
\;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\

\mathbf{elif}\;y \le 2.143447732517704601694518276826181085268 \cdot 10^{-103}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{z - a}, \frac{y}{\frac{1}{z - t}}, x\right)\\

\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 r370842 = x;
        double r370843 = y;
        double r370844 = z;
        double r370845 = t;
        double r370846 = r370844 - r370845;
        double r370847 = a;
        double r370848 = r370844 - r370847;
        double r370849 = r370846 / r370848;
        double r370850 = r370843 * r370849;
        double r370851 = r370842 + r370850;
        return r370851;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r370852 = y;
        double r370853 = -1.7765874073361617e-59;
        bool r370854 = r370852 <= r370853;
        double r370855 = z;
        double r370856 = a;
        double r370857 = r370855 - r370856;
        double r370858 = t;
        double r370859 = r370855 - r370858;
        double r370860 = r370857 / r370859;
        double r370861 = r370852 / r370860;
        double r370862 = x;
        double r370863 = r370861 + r370862;
        double r370864 = 2.1434477325177046e-103;
        bool r370865 = r370852 <= r370864;
        double r370866 = 1.0;
        double r370867 = r370866 / r370857;
        double r370868 = r370866 / r370859;
        double r370869 = r370852 / r370868;
        double r370870 = fma(r370867, r370869, r370862);
        double r370871 = r370859 * r370867;
        double r370872 = fma(r370871, r370852, r370862);
        double r370873 = r370865 ? r370870 : r370872;
        double r370874 = r370854 ? r370863 : r370873;
        return r370874;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target1.2
Herbie0.5
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7765874073361617e-59

    1. Initial program 0.5

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

    2. Simplified0.5

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

    4. Applied clear-num0.5

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

    6. Applied fma-udef0.5

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

    7. Simplified0.4

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

    if -1.7765874073361617e-59 < y < 2.1434477325177046e-103

    1. Initial program 2.5

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

    2. Simplified2.5

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

    4. Applied clear-num2.5

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

    6. Applied fma-udef2.5

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

    7. Simplified2.2

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

    9. Applied div-inv2.3

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

    10. Applied *-un-lft-identity2.3

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

    11. Applied times-frac0.5

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

    12. Applied fma-def0.5

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

    if 2.1434477325177046e-103 < y

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied div-inv0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.776587407336161672094508528940042719675 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{elif}\;y \le 2.143447732517704601694518276826181085268 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z - a}, \frac{y}{\frac{1}{z - t}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{1}{z - a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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)))))