Average Error: 16.6 → 9.2
Time: 21.0s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.25098277828456624542582049460992030006 \cdot 10^{155}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t \le 1694021173610034534209814528:\\ \;\;\;\;y \cdot \frac{t - z}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\

\mathbf{elif}\;t \le 1694021173610034534209814528:\\
\;\;\;\;y \cdot \frac{t - z}{a - t} + \left(y + x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r23271513 = x;
        double r23271514 = y;
        double r23271515 = r23271513 + r23271514;
        double r23271516 = z;
        double r23271517 = t;
        double r23271518 = r23271516 - r23271517;
        double r23271519 = r23271518 * r23271514;
        double r23271520 = a;
        double r23271521 = r23271520 - r23271517;
        double r23271522 = r23271519 / r23271521;
        double r23271523 = r23271515 - r23271522;
        return r23271523;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23271524 = t;
        double r23271525 = -7.250982778284566e+155;
        bool r23271526 = r23271524 <= r23271525;
        double r23271527 = y;
        double r23271528 = z;
        double r23271529 = r23271528 / r23271524;
        double r23271530 = x;
        double r23271531 = fma(r23271527, r23271529, r23271530);
        double r23271532 = 1.6940211736100345e+27;
        bool r23271533 = r23271524 <= r23271532;
        double r23271534 = r23271524 - r23271528;
        double r23271535 = a;
        double r23271536 = r23271535 - r23271524;
        double r23271537 = r23271534 / r23271536;
        double r23271538 = r23271527 * r23271537;
        double r23271539 = r23271527 + r23271530;
        double r23271540 = r23271538 + r23271539;
        double r23271541 = r23271533 ? r23271540 : r23271531;
        double r23271542 = r23271526 ? r23271531 : r23271541;
        return r23271542;
}

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. Taylor expanded around inf 17.9

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

    7. Simplified12.9

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

    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 fma-udef7.0

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

  4. Final simplification9.2

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

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))))