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

Average Error: 22.3 → 0.2

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 512264646.354484260082244873046875\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

C

double f(double x, double y) {
        double r676850 = 1.0;
        double r676851 = x;
        double r676852 = r676850 - r676851;
        double r676853 = y;
        double r676854 = r676852 * r676853;
        double r676855 = r676853 + r676850;
        double r676856 = r676854 / r676855;
        double r676857 = r676850 - r676856;
        return r676857;
}

↓

double f(double x, double y) {
        double r676858 = y;
        double r676859 = -118579514.58451813;
        bool r676860 = r676858 <= r676859;
        double r676861 = 512264646.35448426;
        bool r676862 = r676858 <= r676861;
        double r676863 = !r676862;
        bool r676864 = r676860 || r676863;
        double r676865 = 1.0;
        double r676866 = 1.0;
        double r676867 = r676866 / r676858;
        double r676868 = x;
        double r676869 = r676868 / r676858;
        double r676870 = r676867 - r676869;
        double r676871 = r676865 * r676870;
        double r676872 = r676871 + r676868;
        double r676873 = r676865 - r676868;
        double r676874 = r676858 + r676865;
        double r676875 = r676858 / r676874;
        double r676876 = r676873 * r676875;
        double r676877 = r676865 - r676876;
        double r676878 = r676864 ? r676872 : r676877;
        return r676878;
}

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 < -118579514.58451813 or 512264646.35448426 < y

   1. Initial program 45.4

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

   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 -118579514.58451813 < y < 512264646.35448426

   1. Initial program 0.3

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

   3. Applied *-un-lft-identity 0.3

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -118579514.584518134593963623046875 \lor \neg \left(y \le 512264646.354484260082244873046875\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))