Average Error: 16.6 → 9.2
Time: 18.8s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.25098277828456624542582049460992030006 \cdot 10^{155} \lor \neg \left(t \le 1694021173610034534209814528\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a - t} + \left(y + x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -7.25098277828456624542582049460992030006 \cdot 10^{155} \lor \neg \left(t \le 1694021173610034534209814528\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r373314 = x;
        double r373315 = y;
        double r373316 = r373314 + r373315;
        double r373317 = z;
        double r373318 = t;
        double r373319 = r373317 - r373318;
        double r373320 = r373319 * r373315;
        double r373321 = a;
        double r373322 = r373321 - r373318;
        double r373323 = r373320 / r373322;
        double r373324 = r373316 - r373323;
        return r373324;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r373325 = t;
        double r373326 = -7.250982778284566e+155;
        bool r373327 = r373325 <= r373326;
        double r373328 = 1.6940211736100345e+27;
        bool r373329 = r373325 <= r373328;
        double r373330 = !r373329;
        bool r373331 = r373327 || r373330;
        double r373332 = z;
        double r373333 = r373332 / r373325;
        double r373334 = y;
        double r373335 = x;
        double r373336 = fma(r373333, r373334, r373335);
        double r373337 = r373325 - r373332;
        double r373338 = a;
        double r373339 = r373338 - r373325;
        double r373340 = r373337 / r373339;
        double r373341 = r373334 * r373340;
        double r373342 = r373334 + r373335;
        double r373343 = r373341 + r373342;
        double r373344 = r373331 ? r373336 : r373343;
        return r373344;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.6
Target8.7
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -7.250982778284566e+155 or 1.6940211736100345e+27 < t

    1. Initial program 29.1

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

    2. Simplified19.9

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

    4. Applied add-cube-cbrt20.1

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

    5. Applied associate-/l*20.1

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

    7. Applied fma-udef20.1

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

    8. Simplified20.0

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

    10. Applied pow120.0

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

    11. Applied pow120.0

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

    12. Applied pow-prod-down20.0

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

    13. Simplified29.1

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

    14. Taylor expanded around inf 17.9

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

    15. Simplified12.9

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

    if -7.250982778284566e+155 < t < 1.6940211736100345e+27

    1. Initial program 9.1

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

    2. Simplified7.0

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

    4. Applied add-cube-cbrt7.2

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

    5. Applied associate-/l*7.2

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

    7. Applied fma-udef7.2

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

    8. Simplified7.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.25098277828456624542582049460992030006 \cdot 10^{155} \lor \neg \left(t \le 1694021173610034534209814528\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a - t} + \left(y + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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