Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.2 → 1.5

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \le 3.2897350745799706 \cdot 10^{215}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r634449 = x;
        double r634450 = y;
        double r634451 = r634450 - r634449;
        double r634452 = z;
        double r634453 = t;
        double r634454 = r634452 / r634453;
        double r634455 = r634451 * r634454;
        double r634456 = r634449 + r634455;
        return r634456;
}

↓

double f(double x, double y, double z, double t) {
        double r634457 = z;
        double r634458 = t;
        double r634459 = r634457 / r634458;
        double r634460 = 3.2897350745799706e+215;
        bool r634461 = r634459 <= r634460;
        double r634462 = y;
        double r634463 = x;
        double r634464 = r634462 - r634463;
        double r634465 = fma(r634464, r634459, r634463);
        double r634466 = r634464 * r634457;
        double r634467 = r634466 / r634458;
        double r634468 = r634463 + r634467;
        double r634469 = r634461 ? r634465 : r634468;
        return r634469;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.2
Target	2.5
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ z t) < 3.2897350745799706e+215

   1. Initial program 1.5

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

   2. Simplified 1.5

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

   if 3.2897350745799706e+215 < (/ z t)

   1. Initial program 25.9

      \[x + \left(y - x\right) \cdot \frac{z}{t}\]
   2. Using strategy rm

   3. Applied associate-*r/ 0.5

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \le 3.2897350745799706 \cdot 10^{215}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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