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

Average Error: 7.5 → 0.2

Time: 11.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -8.7834791853338754 \cdot 10^{-11} \lor \neg \left(y \le 3.1266093299729342 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r297742 = x;
        double r297743 = y;
        double r297744 = r297742 + r297743;
        double r297745 = 1.0;
        double r297746 = z;
        double r297747 = r297743 / r297746;
        double r297748 = r297745 - r297747;
        double r297749 = r297744 / r297748;
        return r297749;
}

↓

double f(double x, double y, double z) {
        double r297750 = y;
        double r297751 = -8.783479185333875e-11;
        bool r297752 = r297750 <= r297751;
        double r297753 = 3.126609329972934e-31;
        bool r297754 = r297750 <= r297753;
        double r297755 = !r297754;
        bool r297756 = r297752 || r297755;
        double r297757 = 1.0;
        double r297758 = 1.0;
        double r297759 = x;
        double r297760 = r297759 + r297750;
        double r297761 = r297758 / r297760;
        double r297762 = r297750 / r297760;
        double r297763 = z;
        double r297764 = r297762 / r297763;
        double r297765 = r297761 - r297764;
        double r297766 = r297757 / r297765;
        double r297767 = r297750 / r297763;
        double r297768 = r297758 - r297767;
        double r297769 = r297760 / r297768;
        double r297770 = r297756 ? r297766 : r297769;
        return r297770;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	3.9
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -8.783479185333875e-11 or 3.126609329972934e-31 < y

   1. Initial program 14.0

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

   3. Applied clear-num 14.1

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

   5. Applied div-sub 14.1

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

   6. Simplified 0.3

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

   if -8.783479185333875e-11 < y < 3.126609329972934e-31

   1. Initial program 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.7834791853338754 \cdot 10^{-11} \lor \neg \left(y \le 3.1266093299729342 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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