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

Average Error: 7.6 → 1.5

Time: 14.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r366944 = x;
        double r366945 = y;
        double r366946 = r366944 + r366945;
        double r366947 = 1.0;
        double r366948 = z;
        double r366949 = r366945 / r366948;
        double r366950 = r366947 - r366949;
        double r366951 = r366946 / r366950;
        return r366951;
}

↓

double f(double x, double y, double z) {
        double r366952 = 1.0;
        double r366953 = 1.0;
        double r366954 = x;
        double r366955 = y;
        double r366956 = r366954 + r366955;
        double r366957 = r366953 / r366956;
        double r366958 = r366955 / r366956;
        double r366959 = z;
        double r366960 = r366958 / r366959;
        double r366961 = r366957 - r366960;
        double r366962 = r366952 / r366961;
        return r366962;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	4.1
Herbie	1.5

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

Derivation

1. Initial program 7.6

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

3. Applied clear-num 7.8

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

5. Applied div-sub 7.8

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

6. Simplified 6.7

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

8. Applied associate-/r* 1.5

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

9. Final simplification 1.5

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

Reproduce

herbie shell --seed 2019322 
(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))))