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

Average Error: 7.7 → 3.6

Time: 4.7s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -9.4770144542935185 \cdot 10^{223}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -8.7852555657273445 \cdot 10^{130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \left(\sqrt[3]{\frac{x}{t \cdot z - x}} \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}\right) \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}}{x + 1}\\ \mathbf{elif}\;z \le -2.6731385595801352 \cdot 10^{50}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 4.5656430069255667 \cdot 10^{185}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r812712 = x;
        double r812713 = y;
        double r812714 = z;
        double r812715 = r812713 * r812714;
        double r812716 = r812715 - r812712;
        double r812717 = t;
        double r812718 = r812717 * r812714;
        double r812719 = r812718 - r812712;
        double r812720 = r812716 / r812719;
        double r812721 = r812712 + r812720;
        double r812722 = 1.0;
        double r812723 = r812712 + r812722;
        double r812724 = r812721 / r812723;
        return r812724;
}

↓

double f(double x, double y, double z, double t) {
        double r812725 = z;
        double r812726 = -9.477014454293518e+223;
        bool r812727 = r812725 <= r812726;
        double r812728 = x;
        double r812729 = y;
        double r812730 = t;
        double r812731 = r812729 / r812730;
        double r812732 = r812728 + r812731;
        double r812733 = 1.0;
        double r812734 = r812728 + r812733;
        double r812735 = r812732 / r812734;
        double r812736 = -8.785255565727344e+130;
        bool r812737 = r812725 <= r812736;
        double r812738 = r812730 * r812725;
        double r812739 = r812738 - r812728;
        double r812740 = r812729 / r812739;
        double r812741 = fma(r812740, r812725, r812728);
        double r812742 = r812728 / r812739;
        double r812743 = cbrt(r812742);
        double r812744 = r812743 * r812743;
        double r812745 = r812744 * r812743;
        double r812746 = r812741 - r812745;
        double r812747 = r812746 / r812734;
        double r812748 = -2.6731385595801352e+50;
        bool r812749 = r812725 <= r812748;
        double r812750 = 4.5656430069255667e+185;
        bool r812751 = r812725 <= r812750;
        double r812752 = r812729 * r812725;
        double r812753 = r812752 - r812728;
        double r812754 = 1.0;
        double r812755 = r812754 / r812739;
        double r812756 = r812753 * r812755;
        double r812757 = r812728 + r812756;
        double r812758 = r812757 / r812734;
        double r812759 = r812751 ? r812758 : r812735;
        double r812760 = r812749 ? r812735 : r812759;
        double r812761 = r812737 ? r812747 : r812760;
        double r812762 = r812727 ? r812735 : r812761;
        return r812762;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.7
Target	0.4
Herbie	3.6

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

Derivation

1. Split input into 3 regimes

2. if z < -9.477014454293518e+223 or -8.785255565727344e+130 < z < -2.6731385595801352e+50 or 4.5656430069255667e+185 < z

   1. Initial program 20.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 6.3

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

   if -9.477014454293518e+223 < z < -8.785255565727344e+130

   1. Initial program 18.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 18.0

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

   4. Applied associate-+r- 18.0

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

   5. Simplified 7.1

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

   7. Applied add-cube-cbrt 7.1

      \[\leadsto \frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \color{blue}{\left(\sqrt[3]{\frac{x}{t \cdot z - x}} \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}\right) \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}}}{x + 1}\]

   if -2.6731385595801352e+50 < z < 4.5656430069255667e+185

   1. Initial program 2.2

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-inv 2.3

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.4770144542935185 \cdot 10^{223}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le -8.7852555657273445 \cdot 10^{130}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \left(\sqrt[3]{\frac{x}{t \cdot z - x}} \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}\right) \cdot \sqrt[3]{\frac{x}{t \cdot z - x}}}{x + 1}\\ \mathbf{elif}\;z \le -2.6731385595801352 \cdot 10^{50}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;z \le 4.5656430069255667 \cdot 10^{185}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

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