Average Error: 1.4 → 2.0
Time: 19.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.980533944371433 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right) + x\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -6.980533944371433 \cdot 10^{-281}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r28527134 = x;
        double r28527135 = y;
        double r28527136 = z;
        double r28527137 = t;
        double r28527138 = r28527136 - r28527137;
        double r28527139 = a;
        double r28527140 = r28527139 - r28527137;
        double r28527141 = r28527138 / r28527140;
        double r28527142 = r28527135 * r28527141;
        double r28527143 = r28527134 + r28527142;
        return r28527143;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r28527144 = t;
        double r28527145 = -6.980533944371433e-281;
        bool r28527146 = r28527144 <= r28527145;
        double r28527147 = z;
        double r28527148 = r28527147 - r28527144;
        double r28527149 = a;
        double r28527150 = r28527149 - r28527144;
        double r28527151 = r28527148 / r28527150;
        double r28527152 = y;
        double r28527153 = x;
        double r28527154 = fma(r28527151, r28527152, r28527153);
        double r28527155 = r28527152 / r28527150;
        double r28527156 = r28527155 * r28527148;
        double r28527157 = r28527156 + r28527153;
        double r28527158 = r28527146 ? r28527154 : r28527157;
        return r28527158;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.4
Target0.4
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -6.980533944371433e-281

    1. Initial program 1.1

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

    2. Simplified1.1

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

    if -6.980533944371433e-281 < t

    1. Initial program 1.6

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

    2. Simplified1.6

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

    4. Applied clear-num1.6

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

    6. Applied fma-udef1.6

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

    7. Simplified2.7

      \[\leadsto \color{blue}{\frac{y}{a - t} \cdot \left(z - t\right)} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.980533944371433 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - t} \cdot \left(z - t\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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