Average Error: 16.5 → 8.7
Time: 4.3s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.12467982442109356 \cdot 10^{126} \lor \neg \left(t \le 6.1085637173640515 \cdot 10^{121}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{y}}, t - z, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.12467982442109356 \cdot 10^{126} \lor \neg \left(t \le 6.1085637173640515 \cdot 10^{121}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{y}}, t - z, x + y\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r642322 = x;
        double r642323 = y;
        double r642324 = r642322 + r642323;
        double r642325 = z;
        double r642326 = t;
        double r642327 = r642325 - r642326;
        double r642328 = r642327 * r642323;
        double r642329 = a;
        double r642330 = r642329 - r642326;
        double r642331 = r642328 / r642330;
        double r642332 = r642324 - r642331;
        return r642332;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r642333 = t;
        double r642334 = -1.1246798244210936e+126;
        bool r642335 = r642333 <= r642334;
        double r642336 = 6.1085637173640515e+121;
        bool r642337 = r642333 <= r642336;
        double r642338 = !r642337;
        bool r642339 = r642335 || r642338;
        double r642340 = z;
        double r642341 = r642340 / r642333;
        double r642342 = y;
        double r642343 = x;
        double r642344 = fma(r642341, r642342, r642343);
        double r642345 = 1.0;
        double r642346 = a;
        double r642347 = r642346 - r642333;
        double r642348 = r642347 / r642342;
        double r642349 = r642345 / r642348;
        double r642350 = r642333 - r642340;
        double r642351 = r642343 + r642342;
        double r642352 = fma(r642349, r642350, r642351);
        double r642353 = r642339 ? r642344 : r642352;
        return r642353;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.5
Target8.4
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.1246798244210936e+126 or 6.1085637173640515e+121 < t

    1. Initial program 30.8

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

    2. Simplified22.6

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

    4. Applied div-inv22.6

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

    6. Applied fma-udef22.7

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

    7. Simplified22.6

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

    8. Taylor expanded around inf 16.9

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

    9. Simplified11.8

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

    if -1.1246798244210936e+126 < t < 6.1085637173640515e+121

    1. Initial program 9.5

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

    2. Simplified7.1

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

    4. Applied clear-num7.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a - t}{y}}}, t - z, x + y\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.12467982442109356 \cdot 10^{126} \lor \neg \left(t \le 6.1085637173640515 \cdot 10^{121}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{y}}, t - z, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))