Average Error: 16.9 → 10.2
Time: 5.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.23331323304797557 \cdot 10^{-168} \lor \neg \left(a \le 1.5582813309178771 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;a \le -4.23331323304797557 \cdot 10^{-168} \lor \neg \left(a \le 1.5582813309178771 \cdot 10^{-23}\right):\\
\;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r503476 = x;
        double r503477 = y;
        double r503478 = r503476 + r503477;
        double r503479 = z;
        double r503480 = t;
        double r503481 = r503479 - r503480;
        double r503482 = r503481 * r503477;
        double r503483 = a;
        double r503484 = r503483 - r503480;
        double r503485 = r503482 / r503484;
        double r503486 = r503478 - r503485;
        return r503486;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r503487 = a;
        double r503488 = -4.2333132330479756e-168;
        bool r503489 = r503487 <= r503488;
        double r503490 = 1.558281330917877e-23;
        bool r503491 = r503487 <= r503490;
        double r503492 = !r503491;
        bool r503493 = r503489 || r503492;
        double r503494 = y;
        double r503495 = 1.0;
        double r503496 = t;
        double r503497 = r503487 - r503496;
        double r503498 = r503495 / r503497;
        double r503499 = r503494 * r503498;
        double r503500 = z;
        double r503501 = r503496 - r503500;
        double r503502 = x;
        double r503503 = r503502 + r503494;
        double r503504 = fma(r503499, r503501, r503503);
        double r503505 = r503500 / r503496;
        double r503506 = fma(r503505, r503494, r503502);
        double r503507 = r503493 ? r503504 : r503506;
        return r503507;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.9
Target8.4
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -4.2333132330479756e-168 or 1.558281330917877e-23 < a

    1. Initial program 15.3

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

    2. Simplified9.1

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

    4. Applied div-inv9.1

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

    if -4.2333132330479756e-168 < a < 1.558281330917877e-23

    1. Initial program 20.2

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

    2. Simplified19.4

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

    4. Applied fma-udef19.5

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

    6. Applied div-inv19.5

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

    7. Applied associate-*l*19.3

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

    8. Simplified19.2

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

    9. Taylor expanded around inf 14.0

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

    10. Simplified12.5

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.23331323304797557 \cdot 10^{-168} \lor \neg \left(a \le 1.5582813309178771 \cdot 10^{-23}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot \frac{1}{a - t}, t - z, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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