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

Average Error: 22.3 → 0.2

Time: 14.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4201198044259844344621563904 \lor \neg \left(y \le \frac{2113735014442253}{4}\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{{y}^{3} \cdot \left(1 - x\right)}{{y}^{3} + {1}^{3}}\right)}^{1}\right) - \left(\left(1 - x\right) \cdot \frac{y}{{y}^{3} + {1}^{3}}\right) \cdot \left(1 \cdot 1 - y \cdot 1\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r458313 = 1.0;
        double r458314 = x;
        double r458315 = r458313 - r458314;
        double r458316 = y;
        double r458317 = r458315 * r458316;
        double r458318 = r458316 + r458313;
        double r458319 = r458317 / r458318;
        double r458320 = r458313 - r458319;
        return r458320;
}

↓

double f(double x, double y) {
        double r458321 = y;
        double r458322 = -4.2011980442598443e+27;
        bool r458323 = r458321 <= r458322;
        double r458324 = 2113735014442253.0;
        double r458325 = 4.0;
        double r458326 = r458324 / r458325;
        bool r458327 = r458321 <= r458326;
        double r458328 = !r458327;
        bool r458329 = r458323 || r458328;
        double r458330 = 1.0;
        double r458331 = 1.0;
        double r458332 = r458331 / r458321;
        double r458333 = x;
        double r458334 = r458333 / r458321;
        double r458335 = r458332 - r458334;
        double r458336 = r458330 * r458335;
        double r458337 = r458336 + r458333;
        double r458338 = 3.0;
        double r458339 = pow(r458321, r458338);
        double r458340 = r458330 - r458333;
        double r458341 = r458339 * r458340;
        double r458342 = pow(r458330, r458338);
        double r458343 = r458339 + r458342;
        double r458344 = r458341 / r458343;
        double r458345 = pow(r458344, r458331);
        double r458346 = r458330 - r458345;
        double r458347 = r458321 / r458343;
        double r458348 = r458340 * r458347;
        double r458349 = r458330 * r458330;
        double r458350 = r458321 * r458330;
        double r458351 = r458349 - r458350;
        double r458352 = r458348 * r458351;
        double r458353 = r458346 - r458352;
        double r458354 = r458329 ? r458337 : r458353;
        return r458354;
}

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 < -4.2011980442598443e+27 or 528433753610563.25 < y

   1. Initial program 46.8

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -4.2011980442598443e+27 < y < 528433753610563.25

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity 1.1

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

   4. Applied times-frac 1.0

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

   5. Simplified 1.0

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

   7. Applied flip3-+ 1.1

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

   8. Applied associate-/r/ 1.1

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

   9. Applied associate-*r* 1.1

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

   11. Applied distribute-lft-in 1.1

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

   12. Applied associate--r+ 0.5

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

   13. Simplified 0.5

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

   15. Applied pow1 0.5

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

   16. Applied pow1 0.5

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

   17. Applied pow-prod-down 0.5

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

   18. Applied pow1 0.5

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

   19. Applied pow1 0.5

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

   20. Applied pow-prod-down 0.5

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

   21. Applied pow-prod-down 0.5

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

   22. Simplified 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4201198044259844344621563904 \lor \neg \left(y \le \frac{2113735014442253}{4}\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - {\left(\frac{{y}^{3} \cdot \left(1 - x\right)}{{y}^{3} + {1}^{3}}\right)}^{1}\right) - \left(\left(1 - x\right) \cdot \frac{y}{{y}^{3} + {1}^{3}}\right) \cdot \left(1 \cdot 1 - y \cdot 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))