Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.3 → 0.2

Time: 10.4s

Precision: 64

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -102078540239.528106689453125 \lor \neg \left(y \le 181209685.5427190363407135009765625\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

C

double f(double x, double y) {
        double r585487 = 1.0;
        double r585488 = x;
        double r585489 = r585487 - r585488;
        double r585490 = y;
        double r585491 = r585489 * r585490;
        double r585492 = r585490 + r585487;
        double r585493 = r585491 / r585492;
        double r585494 = r585487 - r585493;
        return r585494;
}

↓

double f(double x, double y) {
        double r585495 = y;
        double r585496 = -102078540239.5281;
        bool r585497 = r585495 <= r585496;
        double r585498 = 181209685.54271904;
        bool r585499 = r585495 <= r585498;
        double r585500 = !r585499;
        bool r585501 = r585497 || r585500;
        double r585502 = 1.0;
        double r585503 = 1.0;
        double r585504 = r585503 / r585495;
        double r585505 = x;
        double r585506 = r585505 / r585495;
        double r585507 = r585504 - r585506;
        double r585508 = r585502 * r585507;
        double r585509 = r585508 + r585505;
        double r585510 = r585502 - r585505;
        double r585511 = r585495 + r585502;
        double r585512 = r585495 / r585511;
        double r585513 = r585510 * r585512;
        double r585514 = r585502 - r585513;
        double r585515 = r585501 ? r585509 : r585514;
        return r585515;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.3
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.848278829724677052581682801246643066:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891002655029296875:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -102078540239.5281 or 181209685.54271904 < y

   1. Initial program 45.7

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

   if -102078540239.5281 < y < 181209685.54271904

   1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.2

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

   4. Applied times-frac 0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{1} \cdot \frac{y}{y + 1}}\]

   5. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -102078540239.528106689453125 \lor \neg \left(y \le 181209685.5427190363407135009765625\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.84827882972468) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891003) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))