Average Error: 1.3 → 0.7
Time: 20.9s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \le -7.558343828768458999521828437649306587911 \cdot 10^{262}:\\ \;\;\;\;\left(z - t\right) \cdot \left(y \cdot \frac{1}{a - t}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z - t}{a - t} \le -7.558343828768458999521828437649306587911 \cdot 10^{262}:\\
\;\;\;\;\left(z - t\right) \cdot \left(y \cdot \frac{1}{a - t}\right) + 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 r484394 = x;
        double r484395 = y;
        double r484396 = z;
        double r484397 = t;
        double r484398 = r484396 - r484397;
        double r484399 = a;
        double r484400 = r484399 - r484397;
        double r484401 = r484398 / r484400;
        double r484402 = r484395 * r484401;
        double r484403 = r484394 + r484402;
        return r484403;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r484404 = z;
        double r484405 = t;
        double r484406 = r484404 - r484405;
        double r484407 = a;
        double r484408 = r484407 - r484405;
        double r484409 = r484406 / r484408;
        double r484410 = -7.558343828768459e+262;
        bool r484411 = r484409 <= r484410;
        double r484412 = y;
        double r484413 = 1.0;
        double r484414 = r484413 / r484408;
        double r484415 = r484412 * r484414;
        double r484416 = r484406 * r484415;
        double r484417 = x;
        double r484418 = r484416 + r484417;
        double r484419 = fma(r484409, r484412, r484417);
        double r484420 = r484411 ? r484418 : r484419;
        return r484420;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- z t) (- a t)) < -7.558343828768459e+262

    1. Initial program 34.9

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

    2. Simplified34.9

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

    4. Applied div-inv34.9

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

    6. Applied add-cube-cbrt35.3

      \[\leadsto \mathsf{fma}\left(\left(z - t\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right) \cdot \sqrt[3]{\frac{1}{a - t}}\right)}, y, x\right)\]

    7. Applied associate-*r*35.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(z - t\right) \cdot \left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{1}{a - t}}}, y, x\right)\]
    8. Using strategy rm

    9. Applied fma-udef35.2

      \[\leadsto \color{blue}{\left(\left(\left(z - t\right) \cdot \left(\sqrt[3]{\frac{1}{a - t}} \cdot \sqrt[3]{\frac{1}{a - t}}\right)\right) \cdot \sqrt[3]{\frac{1}{a - t}}\right) \cdot y + x}\]

    10. Simplified0.3

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

    if -7.558343828768459e+262 < (/ (- z t) (- a t))

    1. Initial program 0.7

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

    2. Simplified0.7

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \le -7.558343828768458999521828437649306587911 \cdot 10^{262}:\\ \;\;\;\;\left(z - t\right) \cdot \left(y \cdot \frac{1}{a - t}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :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)))))