Average Error: 1.3 → 1.3
Time: 19.4s
Precision: 64
\[x + y \cdot \frac{z - t}{a - t}\]
↓
\[\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)\]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(\frac{z - t}{a - t}, y, x\right)
double f(double x, double y, double z, double t, double a) {
        double r314559 = x;
        double r314560 = y;
        double r314561 = z;
        double r314562 = t;
        double r314563 = r314561 - r314562;
        double r314564 = a;
        double r314565 = r314564 - r314562;
        double r314566 = r314563 / r314565;
        double r314567 = r314560 * r314566;
        double r314568 = r314559 + r314567;
        return r314568;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r314569 = z;
        double r314570 = t;
        double r314571 = r314569 - r314570;
        double r314572 = a;
        double r314573 = r314572 - r314570;
        double r314574 = r314571 / r314573;
        double r314575 = y;
        double r314576 = x;
        double r314577 = fma(r314574, r314575, r314576);
        return r314577;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.3
Target0.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241069024247453646278348229 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

  1. Initial program 1.3

    \[x + y \cdot \frac{z - t}{a - t}\]

  2. Simplified1.3

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

  3. Final simplification1.3

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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