Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Average Error: 1.2 → 1.2

Time: 4.1s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{a - t}, x\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r630578 = x;
        double r630579 = y;
        double r630580 = z;
        double r630581 = t;
        double r630582 = r630580 - r630581;
        double r630583 = a;
        double r630584 = r630583 - r630581;
        double r630585 = r630582 / r630584;
        double r630586 = r630579 * r630585;
        double r630587 = r630578 + r630586;
        return r630587;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r630588 = y;
        double r630589 = z;
        double r630590 = t;
        double r630591 = r630589 - r630590;
        double r630592 = 1.0;
        double r630593 = a;
        double r630594 = r630593 - r630590;
        double r630595 = r630592 / r630594;
        double r630596 = r630591 * r630595;
        double r630597 = x;
        double r630598 = fma(r630588, r630596, r630597);
        return r630598;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.2
Target	0.4
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;y \lt -8.50808486055124107 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Initial program 1.2

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

2. Simplified 1.2

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

4. Applied div-inv 1.2

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

5. Final simplification 1.2

   \[\leadsto \mathsf{fma}\left(y, \left(z - t\right) \cdot \frac{1}{a - t}, x\right)\]

Reproduce

herbie shell --seed 2020047 +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)))))