Average Error: 1.2 → 0.8
Time: 19.4s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.4067031403702933 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{elif}\;y \le 4.050712264929229 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.4067031403702933 \cdot 10^{-150}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\

\mathbf{elif}\;y \le 4.050712264929229 \cdot 10^{+58}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r27316638 = x;
        double r27316639 = y;
        double r27316640 = z;
        double r27316641 = t;
        double r27316642 = r27316640 - r27316641;
        double r27316643 = a;
        double r27316644 = r27316643 - r27316641;
        double r27316645 = r27316642 / r27316644;
        double r27316646 = r27316639 * r27316645;
        double r27316647 = r27316638 + r27316646;
        return r27316647;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r27316648 = y;
        double r27316649 = -1.4067031403702933e-150;
        bool r27316650 = r27316648 <= r27316649;
        double r27316651 = z;
        double r27316652 = t;
        double r27316653 = r27316651 - r27316652;
        double r27316654 = a;
        double r27316655 = r27316654 - r27316652;
        double r27316656 = r27316653 / r27316655;
        double r27316657 = x;
        double r27316658 = fma(r27316656, r27316648, r27316657);
        double r27316659 = 4.050712264929229e+58;
        bool r27316660 = r27316648 <= r27316659;
        double r27316661 = r27316653 * r27316648;
        double r27316662 = r27316661 / r27316655;
        double r27316663 = r27316662 + r27316657;
        double r27316664 = r27316660 ? r27316663 : r27316658;
        double r27316665 = r27316650 ? r27316658 : r27316664;
        return r27316665;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target0.4
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.4067031403702933e-150 or 4.050712264929229e+58 < y

    1. Initial program 0.7

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

    2. Simplified0.7

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

    if -1.4067031403702933e-150 < y < 4.050712264929229e+58

    1. Initial program 1.9

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

    2. Simplified1.9

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

    4. Applied clear-num1.9

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

    6. Applied fma-udef1.9

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

    7. Simplified3.1

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

    9. Applied associate-*l/0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.4067031403702933 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \mathbf{elif}\;y \le 4.050712264929229 \cdot 10^{+58}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))))