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

Average Error: 7.7 → 0.2

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.858813669165703078390850859773357267061 \cdot 10^{-284} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r453457 = x;
        double r453458 = y;
        double r453459 = r453457 + r453458;
        double r453460 = 1.0;
        double r453461 = z;
        double r453462 = r453458 / r453461;
        double r453463 = r453460 - r453462;
        double r453464 = r453459 / r453463;
        return r453464;
}

↓

double f(double x, double y, double z) {
        double r453465 = x;
        double r453466 = y;
        double r453467 = r453465 + r453466;
        double r453468 = 1.0;
        double r453469 = z;
        double r453470 = r453466 / r453469;
        double r453471 = r453468 - r453470;
        double r453472 = r453467 / r453471;
        double r453473 = -1.858813669165703e-284;
        bool r453474 = r453472 <= r453473;
        double r453475 = -0.0;
        bool r453476 = r453472 <= r453475;
        double r453477 = !r453476;
        bool r453478 = r453474 || r453477;
        double r453479 = 1.0;
        double r453480 = r453468 / r453467;
        double r453481 = r453467 * r453469;
        double r453482 = r453466 / r453481;
        double r453483 = r453480 - r453482;
        double r453484 = r453479 / r453483;
        double r453485 = r453478 ? r453472 : r453484;
        return r453485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	4.2
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 (/ (+ x y) (- 1.0 (/ y z))) < -1.858813669165703e-284 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 4.1

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

   if -1.858813669165703e-284 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

   1. Initial program 56.8

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

   3. Applied clear-num 56.8

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

   5. Applied div-sub 56.8

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

   6. Simplified 1.9

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.858813669165703078390850859773357267061 \cdot 10^{-284} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.74293107626898565e171) (* (/ (+ y x) (- y)) z) (if (< y 3.55346624560867344e168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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