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

Average Error: 22.7 → 0.2

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -360301218.9043175 \lor \neg \left(y \le 149564260.861771911\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r790425 = 1.0;
        double r790426 = x;
        double r790427 = r790425 - r790426;
        double r790428 = y;
        double r790429 = r790427 * r790428;
        double r790430 = r790428 + r790425;
        double r790431 = r790429 / r790430;
        double r790432 = r790425 - r790431;
        return r790432;
}

↓

double f(double x, double y) {
        double r790433 = y;
        double r790434 = -360301218.9043175;
        bool r790435 = r790433 <= r790434;
        double r790436 = 149564260.8617719;
        bool r790437 = r790433 <= r790436;
        double r790438 = !r790437;
        bool r790439 = r790435 || r790438;
        double r790440 = 1.0;
        double r790441 = 1.0;
        double r790442 = r790441 / r790433;
        double r790443 = x;
        double r790444 = r790443 / r790433;
        double r790445 = r790442 - r790444;
        double r790446 = r790440 * r790445;
        double r790447 = r790446 + r790443;
        double r790448 = r790433 + r790440;
        double r790449 = r790441 / r790448;
        double r790450 = r790440 - r790443;
        double r790451 = r790450 * r790433;
        double r790452 = r790449 * r790451;
        double r790453 = r790440 - r790452;
        double r790454 = r790439 ? r790447 : r790453;
        return r790454;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.7
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -360301218.9043175 or 149564260.8617719 < y

   1. Initial program 46.1

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

   if -360301218.9043175 < y < 149564260.8617719

   1. Initial program 0.2

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

   3. Applied associate-/l* 0.2

      \[\leadsto 1 - \color{blue}{\frac{1 - x}{\frac{y + 1}{y}}}\]
   4. Using strategy rm

   5. Applied div-inv 0.2

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

   6. Applied *-un-lft-identity 0.2

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

   7. Applied times-frac 0.2

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

   8. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -360301218.9043175 \lor \neg \left(y \le 149564260.861771911\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{y + 1} \cdot \left(\left(1 - x\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

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