Average Error: 17.1 → 8.9
Time: 6.9s
Precision: 64
\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.179864941832011171396338266480044311801 \cdot 10^{59} \lor \neg \left(t \le 4.135326618072270316718703166262897552955 \cdot 10^{92}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{y}}, t - z, x + y\right)\\ \end{array}\]
\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.179864941832011171396338266480044311801 \cdot 10^{59} \lor \neg \left(t \le 4.135326618072270316718703166262897552955 \cdot 10^{92}\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 r596555 = x;
        double r596556 = y;
        double r596557 = r596555 + r596556;
        double r596558 = z;
        double r596559 = t;
        double r596560 = r596558 - r596559;
        double r596561 = r596560 * r596556;
        double r596562 = a;
        double r596563 = r596562 - r596559;
        double r596564 = r596561 / r596563;
        double r596565 = r596557 - r596564;
        return r596565;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r596566 = t;
        double r596567 = -1.1798649418320112e+59;
        bool r596568 = r596566 <= r596567;
        double r596569 = 4.1353266180722703e+92;
        bool r596570 = r596566 <= r596569;
        double r596571 = !r596570;
        bool r596572 = r596568 || r596571;
        double r596573 = z;
        double r596574 = r596573 / r596566;
        double r596575 = y;
        double r596576 = x;
        double r596577 = fma(r596574, r596575, r596576);
        double r596578 = 1.0;
        double r596579 = a;
        double r596580 = r596579 - r596566;
        double r596581 = r596580 / r596575;
        double r596582 = r596578 / r596581;
        double r596583 = r596566 - r596573;
        double r596584 = r596576 + r596575;
        double r596585 = fma(r596582, r596583, r596584);
        double r596586 = r596572 ? r596577 : r596585;
        return r596586;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original17.1
Target9.0
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.1798649418320112e+59 or 4.1353266180722703e+92 < t

    1. Initial program 30.1

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

    2. Simplified21.2

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

    4. Applied clear-num21.4

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

    6. Applied fma-udef21.4

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

    7. Simplified21.4

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

    8. Taylor expanded around inf 18.1

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

    9. Simplified12.8

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

    if -1.1798649418320112e+59 < t < 4.1353266180722703e+92

    1. Initial program 7.9

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

    2. Simplified6.1

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

    4. Applied clear-num6.1

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.179864941832011171396338266480044311801 \cdot 10^{59} \lor \neg \left(t \le 4.135326618072270316718703166262897552955 \cdot 10^{92}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a - t}{y}}, t - z, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))