Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Average Error: 7.8 → 0.2

Time: 14.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -560684022086480470999040 \lor \neg \left(y \le 5.796785265433715959339233680935876691365 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r361605 = x;
        double r361606 = y;
        double r361607 = r361605 + r361606;
        double r361608 = 1.0;
        double r361609 = z;
        double r361610 = r361606 / r361609;
        double r361611 = r361608 - r361610;
        double r361612 = r361607 / r361611;
        return r361612;
}

↓

double f(double x, double y, double z) {
        double r361613 = y;
        double r361614 = -5.606840220864805e+23;
        bool r361615 = r361613 <= r361614;
        double r361616 = 5.796785265433716e-08;
        bool r361617 = r361613 <= r361616;
        double r361618 = !r361617;
        bool r361619 = r361615 || r361618;
        double r361620 = 1.0;
        double r361621 = 1.0;
        double r361622 = x;
        double r361623 = r361622 + r361613;
        double r361624 = r361621 / r361623;
        double r361625 = r361613 / r361623;
        double r361626 = z;
        double r361627 = r361625 / r361626;
        double r361628 = r361624 - r361627;
        double r361629 = r361620 / r361628;
        double r361630 = r361613 / r361626;
        double r361631 = r361621 - r361630;
        double r361632 = r361623 / r361631;
        double r361633 = r361619 ? r361629 : r361632;
        return r361633;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
Target	4.1
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -5.606840220864805e+23 or 5.796785265433716e-08 < y

   1. Initial program 15.5

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

   3. Applied clear-num 15.7

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

   5. Applied div-sub 15.7

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

   6. Simplified 15.7

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

   7. Simplified 9.7

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

   9. Applied associate-/r* 0.2

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

   10. Simplified 0.2

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

   if -5.606840220864805e+23 < y < 5.796785265433716e-08

   1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -560684022086480470999040 \lor \neg \left(y \le 5.796785265433715959339233680935876691365 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))