Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 10.3 → 0.1

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -159.566156241908544:\\ \;\;\;\;\frac{x}{{\left(\frac{z}{\left(y - z\right) + 1}\right)}^{1}}\\ \mathbf{elif}\;z \le 8.26470253112780802 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \left(\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r719589 = x;
        double r719590 = y;
        double r719591 = z;
        double r719592 = r719590 - r719591;
        double r719593 = 1.0;
        double r719594 = r719592 + r719593;
        double r719595 = r719589 * r719594;
        double r719596 = r719595 / r719591;
        return r719596;
}

↓

double f(double x, double y, double z) {
        double r719597 = z;
        double r719598 = -159.56615624190854;
        bool r719599 = r719597 <= r719598;
        double r719600 = x;
        double r719601 = y;
        double r719602 = r719601 - r719597;
        double r719603 = 1.0;
        double r719604 = r719602 + r719603;
        double r719605 = r719597 / r719604;
        double r719606 = 1.0;
        double r719607 = pow(r719605, r719606);
        double r719608 = r719600 / r719607;
        double r719609 = 8.264702531127808e-08;
        bool r719610 = r719597 <= r719609;
        double r719611 = r719600 / r719597;
        double r719612 = r719611 * r719604;
        double r719613 = r719606 * r719612;
        double r719614 = r719604 / r719597;
        double r719615 = r719600 * r719614;
        double r719616 = r719610 ? r719613 : r719615;
        double r719617 = r719599 ? r719608 : r719616;
        return r719617;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.3
Target	0.4
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -159.56615624190854

   1. Initial program 17.1

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

   3. Applied associate-/l* 0.1

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

   5. Applied pow1 0.1

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

   if -159.56615624190854 < z < 8.264702531127808e-08

   1. Initial program 0.1

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

   3. Applied associate-/l* 8.5

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

   5. Applied pow1 8.5

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

   7. Applied *-un-lft-identity 8.5

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

   8. Applied *-un-lft-identity 8.5

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

   9. Applied times-frac 8.5

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

   10. Applied unpow-prod-down 8.5

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

   11. Applied *-un-lft-identity 8.5

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

   12. Applied times-frac 8.5

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

   13. Simplified 8.5

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

   14. Simplified 0.1

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

   if 8.264702531127808e-08 < z

   1. Initial program 16.3

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

   3. Applied *-un-lft-identity 16.3

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

   4. Applied times-frac 0.1

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

   5. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -159.566156241908544:\\ \;\;\;\;\frac{x}{{\left(\frac{z}{\left(y - z\right) + 1}\right)}^{1}}\\ \mathbf{elif}\;z \le 8.26470253112780802 \cdot 10^{-8}:\\ \;\;\;\;1 \cdot \left(\frac{x}{z} \cdot \left(\left(y - z\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y - z\right) + 1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

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