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

Average Error: 7.9 → 0.3

Time: 9.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -5.260737244167179 \cdot 10^{-286} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r719633 = x;
        double r719634 = y;
        double r719635 = r719633 + r719634;
        double r719636 = 1.0;
        double r719637 = z;
        double r719638 = r719634 / r719637;
        double r719639 = r719636 - r719638;
        double r719640 = r719635 / r719639;
        return r719640;
}

↓

double f(double x, double y, double z) {
        double r719641 = x;
        double r719642 = y;
        double r719643 = r719641 + r719642;
        double r719644 = 1.0;
        double r719645 = z;
        double r719646 = r719642 / r719645;
        double r719647 = r719644 - r719646;
        double r719648 = r719643 / r719647;
        double r719649 = -5.260737244167179e-286;
        bool r719650 = r719648 <= r719649;
        double r719651 = 0.0;
        bool r719652 = r719648 <= r719651;
        double r719653 = !r719652;
        bool r719654 = r719650 || r719653;
        double r719655 = 1.0;
        double r719656 = r719644 / r719643;
        double r719657 = r719643 * r719645;
        double r719658 = r719642 / r719657;
        double r719659 = r719656 - r719658;
        double r719660 = r719655 / r719659;
        double r719661 = r719654 ? r719648 : r719660;
        return r719661;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.9
Target	4.1
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -5.260737244167179e-286 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 0.1

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

   if -5.260737244167179e-286 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

   1. Initial program 58.4

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

   3. Applied clear-num 58.4

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

   5. Applied div-sub 58.4

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

   6. Simplified 1.6

      \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{y}{\left(x + y\right) \cdot z}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -5.260737244167179 \cdot 10^{-286} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))