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

Average Error: 7.5 → 0.3

Time: 3.3s

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 r763055 = x;
        double r763056 = y;
        double r763057 = r763055 + r763056;
        double r763058 = 1.0;
        double r763059 = z;
        double r763060 = r763056 / r763059;
        double r763061 = r763058 - r763060;
        double r763062 = r763057 / r763061;
        return r763062;
}

↓

double f(double x, double y, double z) {
        double r763063 = x;
        double r763064 = y;
        double r763065 = r763063 + r763064;
        double r763066 = 1.0;
        double r763067 = z;
        double r763068 = r763064 / r763067;
        double r763069 = r763066 - r763068;
        double r763070 = r763065 / r763069;
        double r763071 = -6.318248638208635e-275;
        bool r763072 = r763070 <= r763071;
        double r763073 = 0.0;
        bool r763074 = r763070 <= r763073;
        double r763075 = !r763074;
        bool r763076 = r763072 || r763075;
        double r763077 = 1.0;
        double r763078 = r763066 / r763065;
        double r763079 = r763065 * r763067;
        double r763080 = r763064 / r763079;
        double r763081 = r763078 - r763080;
        double r763082 = r763077 / r763081;
        double r763083 = r763076 ? r763070 : r763082;
        return r763083;
}

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 
(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))))