Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Average Error: 28.1 → 1.1

Time: 4.7s

Precision: 64

Math

\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le 1.142862718542397604994653604881790849452 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - {\left(\sqrt[3]{z}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{z}\right)}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\left|z\right|}{\sqrt{y}} \cdot \frac{\left|z\right|}{\sqrt{y}}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r741670 = x;
        double r741671 = r741670 * r741670;
        double r741672 = y;
        double r741673 = r741672 * r741672;
        double r741674 = r741671 + r741673;
        double r741675 = z;
        double r741676 = r741675 * r741675;
        double r741677 = r741674 - r741676;
        double r741678 = 2.0;
        double r741679 = r741672 * r741678;
        double r741680 = r741677 / r741679;
        return r741680;
}

↓

double f(double x, double y, double z) {
        double r741681 = y;
        double r741682 = 1.1428627185424e-311;
        bool r741683 = r741681 <= r741682;
        double r741684 = 0.5;
        double r741685 = x;
        double r741686 = r741681 / r741685;
        double r741687 = r741685 / r741686;
        double r741688 = r741681 + r741687;
        double r741689 = z;
        double r741690 = cbrt(r741689);
        double r741691 = 4.0;
        double r741692 = pow(r741690, r741691);
        double r741693 = 2.0;
        double r741694 = pow(r741690, r741693);
        double r741695 = r741694 / r741681;
        double r741696 = r741692 * r741695;
        double r741697 = r741688 - r741696;
        double r741698 = r741684 * r741697;
        double r741699 = fabs(r741689);
        double r741700 = sqrt(r741681);
        double r741701 = r741699 / r741700;
        double r741702 = r741701 * r741701;
        double r741703 = r741688 - r741702;
        double r741704 = r741684 * r741703;
        double r741705 = r741683 ? r741698 : r741704;
        return r741705;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	28.1
Target	0.2
Herbie	1.1

\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

1. Split input into 2 regimes

2. if y < 1.1428627185424e-311

   1. Initial program 28.3

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

   2. Taylor expanded around 0 12.4

      \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

   3. Simplified 12.4

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
   4. Using strategy rm

   5. Applied unpow2 12.4

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{x \cdot x}}{y}\right) - \frac{{z}^{2}}{y}\right)\]

   6. Applied associate-/l* 6.8

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{x}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y}\right)\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 6.8

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{2}}{\color{blue}{1 \cdot y}}\right)\]

   9. Applied add-cube-cbrt 7.1

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{\color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}}^{2}}{1 \cdot y}\right)\]

   10. Applied unpow-prod-down 7.1

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\color{blue}{{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}^{2} \cdot {\left(\sqrt[3]{z}\right)}^{2}}}{1 \cdot y}\right)\]

   11. Applied times-frac 2.0

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{z}\right)}^{2}}{y}}\right)\]

   12. Simplified 2.0

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{{\left(\sqrt[3]{z}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{z}\right)}^{2}}{y}\right)\]

   if 1.1428627185424e-311 < y

   1. Initial program 28.0

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

   2. Taylor expanded around 0 12.2

      \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

   3. Simplified 12.2

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
   4. Using strategy rm

   5. Applied unpow2 12.2

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{x \cdot x}}{y}\right) - \frac{{z}^{2}}{y}\right)\]

   6. Applied associate-/l* 6.7

      \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{x}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y}\right)\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 6.7

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{2}}{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right)\]

   9. Applied add-sqr-sqrt 6.7

      \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\color{blue}{\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}}}{\sqrt{y} \cdot \sqrt{y}}\right)\]

   10. Applied times-frac 6.7

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{\sqrt{{z}^{2}}}{\sqrt{y}} \cdot \frac{\sqrt{{z}^{2}}}{\sqrt{y}}}\right)\]

   11. Simplified 6.7

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{\left|z\right|}{\sqrt{y}}} \cdot \frac{\sqrt{{z}^{2}}}{\sqrt{y}}\right)\]

   12. Simplified 0.2

       \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\left|z\right|}{\sqrt{y}} \cdot \color{blue}{\frac{\left|z\right|}{\sqrt{y}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.142862718542397604994653604881790849452 \cdot 10^{-311}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - {\left(\sqrt[3]{z}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{z}\right)}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\left|z\right|}{\sqrt{y}} \cdot \frac{\left|z\right|}{\sqrt{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2)))