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

Average Error: 1.4 → 1.5

Time: 26.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0071455594193599 \cdot 10^{-271} \lor \neg \left(t \le 7.46702847266602817 \cdot 10^{-71}\right):\\ \;\;\;\;x + y \cdot \left(\frac{z}{a - t} - \frac{t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r657271 = x;
        double r657272 = y;
        double r657273 = z;
        double r657274 = t;
        double r657275 = r657273 - r657274;
        double r657276 = a;
        double r657277 = r657276 - r657274;
        double r657278 = r657275 / r657277;
        double r657279 = r657272 * r657278;
        double r657280 = r657271 + r657279;
        return r657280;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r657281 = t;
        double r657282 = -1.0071455594193599e-271;
        bool r657283 = r657281 <= r657282;
        double r657284 = 7.467028472666028e-71;
        bool r657285 = r657281 <= r657284;
        double r657286 = !r657285;
        bool r657287 = r657283 || r657286;
        double r657288 = x;
        double r657289 = y;
        double r657290 = z;
        double r657291 = a;
        double r657292 = r657291 - r657281;
        double r657293 = r657290 / r657292;
        double r657294 = r657281 / r657292;
        double r657295 = r657293 - r657294;
        double r657296 = r657289 * r657295;
        double r657297 = r657288 + r657296;
        double r657298 = r657290 - r657281;
        double r657299 = r657289 * r657298;
        double r657300 = 1.0;
        double r657301 = r657300 / r657292;
        double r657302 = r657299 * r657301;
        double r657303 = r657288 + r657302;
        double r657304 = r657287 ? r657297 : r657303;
        return r657304;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.4
Target	0.5
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if t < -1.0071455594193599e-271 or 7.467028472666028e-71 < t

   1. Initial program 0.9

      \[x + y \cdot \frac{z - t}{a - t}\]
   2. Using strategy rm

   3. Applied div-sub 0.8

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

   if -1.0071455594193599e-271 < t < 7.467028472666028e-71

   1. Initial program 3.5

      \[x + y \cdot \frac{z - t}{a - t}\]
   2. Using strategy rm

   3. Applied div-inv 3.5

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

   4. Applied associate-*r* 3.9

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0071455594193599 \cdot 10^{-271} \lor \neg \left(t \le 7.46702847266602817 \cdot 10^{-71}\right):\\ \;\;\;\;x + y \cdot \left(\frac{z}{a - t} - \frac{t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))))