Average Error: 16.4 → 8.8
Time: 48.3s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.056949652602892724761072253462552924838 \cdot 10^{52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 2.835171245151349837107842028674357428893 \cdot 10^{130}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.056949652602892724761072253462552924838 \cdot 10^{52}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\

\mathbf{elif}\;t \le 2.835171245151349837107842028674357428893 \cdot 10^{130}:\\
\;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\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 r25996432 = x;
        double r25996433 = y;
        double r25996434 = r25996432 + r25996433;
        double r25996435 = z;
        double r25996436 = t;
        double r25996437 = r25996435 - r25996436;
        double r25996438 = r25996437 * r25996433;
        double r25996439 = a;
        double r25996440 = r25996439 - r25996436;
        double r25996441 = r25996438 / r25996440;
        double r25996442 = r25996434 - r25996441;
        return r25996442;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25996443 = t;
        double r25996444 = -1.0569496526028927e+52;
        bool r25996445 = r25996443 <= r25996444;
        double r25996446 = z;
        double r25996447 = r25996446 / r25996443;
        double r25996448 = y;
        double r25996449 = x;
        double r25996450 = fma(r25996447, r25996448, r25996449);
        double r25996451 = 2.83517124515135e+130;
        bool r25996452 = r25996443 <= r25996451;
        double r25996453 = r25996443 - r25996446;
        double r25996454 = a;
        double r25996455 = r25996454 - r25996443;
        double r25996456 = r25996448 / r25996455;
        double r25996457 = r25996453 * r25996456;
        double r25996458 = r25996448 + r25996449;
        double r25996459 = r25996457 + r25996458;
        double r25996460 = r25996452 ? r25996459 : r25996450;
        double r25996461 = r25996445 ? r25996450 : r25996460;
        return r25996461;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original16.4
Target8.3
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.0569496526028927e+52 or 2.83517124515135e+130 < t

    1. Initial program 29.5

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

    2. Simplified20.8

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

    3. Taylor expanded around inf 18.1

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

    4. Simplified12.8

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

    if -1.0569496526028927e+52 < t < 2.83517124515135e+130

    1. Initial program 8.4

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

    2. Simplified6.2

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

    4. Applied fma-udef6.3

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.056949652602892724761072253462552924838 \cdot 10^{52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \le 2.835171245151349837107842028674357428893 \cdot 10^{130}:\\ \;\;\;\;\left(t - z\right) \cdot \frac{y}{a - t} + \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))