Average Error: 1.4 → 0.5
Time: 4.5s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -901067.937205575406551361083984375 \lor \neg \left(y \le 5.163724188916300428001985018951606340548 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \end{array}\]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -901067.937205575406551361083984375 \lor \neg \left(y \le 5.163724188916300428001985018951606340548 \cdot 10^{-43}\right):\\
\;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r606517 = x;
        double r606518 = y;
        double r606519 = z;
        double r606520 = t;
        double r606521 = r606519 - r606520;
        double r606522 = a;
        double r606523 = r606519 - r606522;
        double r606524 = r606521 / r606523;
        double r606525 = r606518 * r606524;
        double r606526 = r606517 + r606525;
        return r606526;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r606527 = y;
        double r606528 = -901067.9372055754;
        bool r606529 = r606527 <= r606528;
        double r606530 = 5.1637241889163004e-43;
        bool r606531 = r606527 <= r606530;
        double r606532 = !r606531;
        bool r606533 = r606529 || r606532;
        double r606534 = z;
        double r606535 = a;
        double r606536 = r606534 - r606535;
        double r606537 = t;
        double r606538 = r606534 - r606537;
        double r606539 = r606536 / r606538;
        double r606540 = r606527 / r606539;
        double r606541 = x;
        double r606542 = r606540 + r606541;
        double r606543 = r606527 * r606538;
        double r606544 = r606543 / r606536;
        double r606545 = r606544 + r606541;
        double r606546 = r606533 ? r606542 : r606545;
        return r606546;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.4
Target1.2
Herbie0.5
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -901067.9372055754 or 5.1637241889163004e-43 < y

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied clear-num0.7

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

    6. Applied fma-udef0.7

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

    7. Simplified0.6

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

    if -901067.9372055754 < y < 5.1637241889163004e-43

    1. Initial program 2.1

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

    2. Simplified2.1

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

    4. Applied add-cube-cbrt2.3

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

    6. Applied fma-udef2.3

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

    7. Simplified0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -901067.937205575406551361083984375 \lor \neg \left(y \le 5.163724188916300428001985018951606340548 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))))