Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Error: 12.4 → 12.4

Time: 8.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r691620 = x;
        double r691621 = y;
        double r691622 = z;
        double r691623 = r691621 - r691622;
        double r691624 = r691620 * r691623;
        double r691625 = r691624 / r691621;
        return r691625;
}

↓

double f(double x, double y, double z) {
        double r691626 = x;
        double r691627 = y;
        double r691628 = z;
        double r691629 = r691627 - r691628;
        double r691630 = r691626 * r691629;
        double r691631 = r691630 / r691627;
        return r691631;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.3
Herbie	12.4

\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.290655561206784e+97

   1. Initial program 31.2

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

   3. Applied associate-/l* 0.1

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

   5. Applied clear-num 0.2

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

   7. Applied *-un-lft-identity 0.2

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

   8. Applied *-un-lft-identity 0.2

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

   9. Applied *-un-lft-identity 0.2

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

   10. Applied times-frac 0.2

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

   11. Applied times-frac 0.2

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

   12. Applied add-sqr-sqrt 0.2

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

   13. Applied times-frac 0.2

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

   14. Simplified 0.2

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

   15. Simplified 0.1

       \[\leadsto 1 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{y - z}}}\]
   16. Using strategy rm

   17. Applied div-inv 0.1

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

   18. Simplified 0.1

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

   if -1.290655561206784e+97 < x < 2.7317275713955442e-15

   1. Initial program 5.6

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

   3. Applied associate-/l* 5.2

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

   5. Applied clear-num 5.3

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

   7. Applied *-un-lft-identity 5.3

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

   8. Applied *-un-lft-identity 5.3

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

   9. Applied *-un-lft-identity 5.3

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

   10. Applied times-frac 5.3

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

   11. Applied times-frac 5.3

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

   12. Applied add-sqr-sqrt 5.3

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

   13. Applied times-frac 5.3

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

   14. Simplified 5.3

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

   15. Simplified 5.2

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

   16. Taylor expanded around 0 2.7

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

   if 2.7317275713955442e-15 < x

   1. Initial program 20.8

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

   3. Applied associate-/l* 0.1

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

   5. Applied clear-num 0.2

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

   7. Applied *-un-lft-identity 0.2

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

   8. Applied *-un-lft-identity 0.2

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

   9. Applied *-un-lft-identity 0.2

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

   10. Applied times-frac 0.2

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

   11. Applied times-frac 0.2

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

   12. Applied add-sqr-sqrt 0.2

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

   13. Applied times-frac 0.2

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

   14. Simplified 0.2

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

   15. Simplified 0.1

       \[\leadsto 1 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{y - z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \frac{x \cdot \left(y - z\right)}{y}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e104) (- x (/ (* z x) y)) (if (< z 1.69397660138285259e213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))