Average Error: 1.3 → 1.1
Time: 21.1s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} = -\infty:\\ \;\;\;\;\left(x + y\right) - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t} - \frac{t}{a - t}, y, x\right)\\ \end{array}\]
x + y \cdot \frac{z - t}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z - t}{a - t} = -\infty:\\
\;\;\;\;\left(x + y\right) - \frac{y \cdot z}{t}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r25412484 = x;
        double r25412485 = y;
        double r25412486 = z;
        double r25412487 = t;
        double r25412488 = r25412486 - r25412487;
        double r25412489 = a;
        double r25412490 = r25412489 - r25412487;
        double r25412491 = r25412488 / r25412490;
        double r25412492 = r25412485 * r25412491;
        double r25412493 = r25412484 + r25412492;
        return r25412493;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25412494 = z;
        double r25412495 = t;
        double r25412496 = r25412494 - r25412495;
        double r25412497 = a;
        double r25412498 = r25412497 - r25412495;
        double r25412499 = r25412496 / r25412498;
        double r25412500 = -inf.0;
        bool r25412501 = r25412499 <= r25412500;
        double r25412502 = x;
        double r25412503 = y;
        double r25412504 = r25412502 + r25412503;
        double r25412505 = r25412503 * r25412494;
        double r25412506 = r25412505 / r25412495;
        double r25412507 = r25412504 - r25412506;
        double r25412508 = r25412494 / r25412498;
        double r25412509 = r25412495 / r25412498;
        double r25412510 = r25412508 - r25412509;
        double r25412511 = fma(r25412510, r25412503, r25412502);
        double r25412512 = r25412501 ? r25412507 : r25412511;
        return r25412512;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- z t) (- a t)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around inf 32.4

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

    if -inf.0 < (/ (- z t) (- a t))

    1. Initial program 0.8

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

    2. Simplified0.8

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

    4. Applied div-sub0.8

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2019192 +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)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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