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

Average Error: 7.7 → 7.7

Time: 13.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r403371 = x;
        double r403372 = y;
        double r403373 = r403371 + r403372;
        double r403374 = 1.0;
        double r403375 = z;
        double r403376 = r403372 / r403375;
        double r403377 = r403374 - r403376;
        double r403378 = r403373 / r403377;
        return r403378;
}

↓

double f(double x, double y, double z) {
        double r403379 = x;
        double r403380 = y;
        double r403381 = r403379 + r403380;
        double r403382 = 1.0;
        double r403383 = z;
        double r403384 = r403380 / r403383;
        double r403385 = r403382 - r403384;
        double r403386 = r403381 / r403385;
        return r403386;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.7
Target	3.9
Herbie	7.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.371613530277043e-283 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 4.3

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

   if -1.371613530277043e-283 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

   1. Initial program 56.1

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

   3. Applied add-sqr-sqrt 60.2

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

   4. Applied add-sqr-sqrt 62.7

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

   5. Applied times-frac 62.7

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

   6. Applied add-sqr-sqrt 62.7

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

   7. Applied difference-of-squares 62.7

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

   8. Applied add-sqr-sqrt 62.8

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

   9. Applied times-frac 62.8

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

4. Final simplification 7.7

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

Reproduce

herbie shell --seed 2019291 
(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))))