Average Error: 1.3 → 0.5
Time: 36.3s
Precision: 64
\[x + y \cdot \frac{z - t}{z - a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.15499137832583823 \cdot 10^{-103} \lor \neg \left(y \le 64647305041162224\right):\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \end{array}\]
x + y \cdot \frac{z - t}{z - a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.15499137832583823 \cdot 10^{-103} \lor \neg \left(y \le 64647305041162224\right):\\
\;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r393464 = x;
        double r393465 = y;
        double r393466 = z;
        double r393467 = t;
        double r393468 = r393466 - r393467;
        double r393469 = a;
        double r393470 = r393466 - r393469;
        double r393471 = r393468 / r393470;
        double r393472 = r393465 * r393471;
        double r393473 = r393464 + r393472;
        return r393473;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r393474 = y;
        double r393475 = -2.1549913783258382e-103;
        bool r393476 = r393474 <= r393475;
        double r393477 = 6.464730504116222e+16;
        bool r393478 = r393474 <= r393477;
        double r393479 = !r393478;
        bool r393480 = r393476 || r393479;
        double r393481 = z;
        double r393482 = a;
        double r393483 = r393481 - r393482;
        double r393484 = t;
        double r393485 = r393481 - r393484;
        double r393486 = r393483 / r393485;
        double r393487 = r393474 / r393486;
        double r393488 = x;
        double r393489 = r393487 + r393488;
        double r393490 = r393485 * r393474;
        double r393491 = r393490 / r393483;
        double r393492 = r393491 + r393488;
        double r393493 = r393480 ? r393489 : r393492;
        return r393493;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if y < -2.1549913783258382e-103 or 6.464730504116222e+16 < 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 clear-num0.7

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

    6. Applied fma-udef0.7

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

    7. Simplified0.7

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

    if -2.1549913783258382e-103 < y < 6.464730504116222e+16

    1. Initial program 2.1

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

    2. Simplified2.1

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

    4. Applied div-sub2.1

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

    6. Applied fma-udef2.1

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

    8. Applied sub-div2.1

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

    9. Applied associate-*l/0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.15499137832583823 \cdot 10^{-103} \lor \neg \left(y \le 64647305041162224\right):\\ \;\;\;\;\frac{y}{\frac{z - a}{z - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{z - a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

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

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