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

Average Error: 7.5 → 0.3

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -6.3182486382086351 \cdot 10^{-275} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r577968 = x;
        double r577969 = y;
        double r577970 = r577968 + r577969;
        double r577971 = 1.0;
        double r577972 = z;
        double r577973 = r577969 / r577972;
        double r577974 = r577971 - r577973;
        double r577975 = r577970 / r577974;
        return r577975;
}

↓

double f(double x, double y, double z) {
        double r577976 = x;
        double r577977 = y;
        double r577978 = r577976 + r577977;
        double r577979 = 1.0;
        double r577980 = z;
        double r577981 = r577977 / r577980;
        double r577982 = r577979 - r577981;
        double r577983 = r577978 / r577982;
        double r577984 = -6.318248638208635e-275;
        bool r577985 = r577983 <= r577984;
        double r577986 = 0.0;
        bool r577987 = r577983 <= r577986;
        double r577988 = !r577987;
        bool r577989 = r577985 || r577988;
        double r577990 = 1.0;
        double r577991 = r577979 / r577978;
        double r577992 = r577978 * r577980;
        double r577993 = r577977 / r577992;
        double r577994 = r577991 - r577993;
        double r577995 = r577990 / r577994;
        double r577996 = r577989 ? r577983 : r577995;
        return r577996;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	3.9
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))) < -6.318248638208635e-275 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 0.1

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

   if -6.318248638208635e-275 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

   1. Initial program 57.4

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

   3. Applied clear-num 57.5

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

   5. Applied div-sub 57.5

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

   7. Applied div-inv 57.5

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

   8. Applied associate-/l* 2.0

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

   9. Simplified 2.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -6.3182486382086351 \cdot 10^{-275} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))