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

Average Error: 1.3 → 0.5

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -250454101183023.0625 \lor \neg \left(y \le 1.180553323745699933141041157702311268804 \cdot 10^{-75}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r583107 = x;
        double r583108 = y;
        double r583109 = z;
        double r583110 = t;
        double r583111 = r583109 - r583110;
        double r583112 = a;
        double r583113 = r583112 - r583110;
        double r583114 = r583111 / r583113;
        double r583115 = r583108 * r583114;
        double r583116 = r583107 + r583115;
        return r583116;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r583117 = y;
        double r583118 = -250454101183023.06;
        bool r583119 = r583117 <= r583118;
        double r583120 = 1.1805533237457e-75;
        bool r583121 = r583117 <= r583120;
        double r583122 = !r583121;
        bool r583123 = r583119 || r583122;
        double r583124 = x;
        double r583125 = z;
        double r583126 = t;
        double r583127 = r583125 - r583126;
        double r583128 = a;
        double r583129 = r583128 - r583126;
        double r583130 = r583127 / r583129;
        double r583131 = r583117 * r583130;
        double r583132 = r583124 + r583131;
        double r583133 = r583127 * r583117;
        double r583134 = r583133 / r583129;
        double r583135 = r583124 + r583134;
        double r583136 = r583123 ? r583132 : r583135;
        return r583136;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.3
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if y < -250454101183023.06 or 1.1805533237457e-75 < y

   1. Initial program 0.6

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

   if -250454101183023.06 < y < 1.1805533237457e-75

   1. Initial program 2.0

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

   3. Applied div-inv 2.1

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

   5. Applied un-div-inv 2.0

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

   6. Applied associate-*r/ 0.4

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

   7. Simplified 0.4

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -250454101183023.0625 \lor \neg \left(y \le 1.180553323745699933141041157702311268804 \cdot 10^{-75}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))))