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

Average Error: 22.8 → 0.1

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2868269604.961124897003173828125 \lor \neg \left(y \le 186543853.3156950771808624267578125\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

C

double f(double x, double y) {
        double r584288 = 1.0;
        double r584289 = x;
        double r584290 = r584288 - r584289;
        double r584291 = y;
        double r584292 = r584290 * r584291;
        double r584293 = r584291 + r584288;
        double r584294 = r584292 / r584293;
        double r584295 = r584288 - r584294;
        return r584295;
}

↓

double f(double x, double y) {
        double r584296 = y;
        double r584297 = -2868269604.961125;
        bool r584298 = r584296 <= r584297;
        double r584299 = 186543853.31569508;
        bool r584300 = r584296 <= r584299;
        double r584301 = !r584300;
        bool r584302 = r584298 || r584301;
        double r584303 = x;
        double r584304 = 1.0;
        double r584305 = r584304 / r584296;
        double r584306 = r584303 / r584296;
        double r584307 = r584304 * r584306;
        double r584308 = r584305 - r584307;
        double r584309 = r584303 + r584308;
        double r584310 = r584304 - r584303;
        double r584311 = r584296 + r584304;
        double r584312 = r584296 / r584311;
        double r584313 = r584310 * r584312;
        double r584314 = r584304 - r584313;
        double r584315 = r584302 ? r584309 : r584314;
        return r584315;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.8
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -2868269604.961125 or 186543853.31569508 < y

   1. Initial program 46.9

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

   3. Applied *-un-lft-identity 46.9

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

   4. Applied times-frac 30.3

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

   5. Simplified 30.3

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

   7. Applied flip3-- 51.3

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{1 \cdot 1 + \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) + 1 \cdot \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)}}\]

   8. Simplified 51.3

      \[\leadsto \frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)}}\]
   9. Using strategy rm

   10. Applied flip-+ 54.1

       \[\leadsto \frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right) \cdot \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)}{1 \cdot 1 - \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)}}}\]

   11. Applied associate-/r/ 54.1

       \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right) \cdot \left(\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)} \cdot \left(1 \cdot 1 - \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)}\]

   12. Simplified 54.9

       \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right)}^{3}}{1 \cdot {1}^{3} - {\left(\frac{\left(1 - x\right) \cdot y}{y + 1}\right)}^{\left(2 \cdot 1\right)} \cdot \left(\left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)}} \cdot \left(1 \cdot 1 - \left(\left(1 - x\right) \cdot \frac{y}{y + 1}\right) \cdot \left(1 + \left(1 - x\right) \cdot \frac{y}{y + 1}\right)\right)\]

   13. Taylor expanded around inf 0.1

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

   14. Simplified 0.1

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

   if -2868269604.961125 < y < 186543853.31569508

   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 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2868269604.961124897003173828125 \lor \neg \left(y \le 186543853.3156950771808624267578125\right):\\ \;\;\;\;x + \left(\frac{1}{y} - 1 \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{y + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))