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

Average Error: 14.0 → 2.3

Time: 3.4s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.45613981804384912 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -5.31650938160314506 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 3.3409790596422747 \cdot 10^{-221}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.8490367773301062 \cdot 10^{81}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r633191 = x;
        double r633192 = y;
        double r633193 = z;
        double r633194 = r633192 / r633193;
        double r633195 = t;
        double r633196 = r633194 * r633195;
        double r633197 = r633196 / r633195;
        double r633198 = r633191 * r633197;
        return r633198;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r633199 = y;
        double r633200 = z;
        double r633201 = r633199 / r633200;
        double r633202 = -1.456139818043849e+278;
        bool r633203 = r633201 <= r633202;
        double r633204 = x;
        double r633205 = r633204 * r633199;
        double r633206 = r633205 / r633200;
        double r633207 = -5.316509381603145e-21;
        bool r633208 = r633201 <= r633207;
        double r633209 = r633204 * r633201;
        double r633210 = 3.3409790596422747e-221;
        bool r633211 = r633201 <= r633210;
        double r633212 = 7.849036777330106e+81;
        bool r633213 = r633201 <= r633212;
        double r633214 = r633200 / r633199;
        double r633215 = r633204 / r633214;
        double r633216 = r633213 ? r633215 : r633206;
        double r633217 = r633211 ? r633206 : r633216;
        double r633218 = r633208 ? r633209 : r633217;
        double r633219 = r633203 ? r633206 : r633218;
        return r633219;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.0
Target	1.5
Herbie	2.3

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.20672205123045005 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390633 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.65895442315341522 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.0087180502407133 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -1.456139818043849e+278 or -5.316509381603145e-21 < (/ y z) < 3.3409790596422747e-221 or 7.849036777330106e+81 < (/ y z)

   1. Initial program 18.6

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 10.7

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 3.9

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

   if -1.456139818043849e+278 < (/ y z) < -5.316509381603145e-21

   1. Initial program 10.4

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 0.3

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

   if 3.3409790596422747e-221 < (/ y z) < 7.849036777330106e+81

   1. Initial program 6.3

      \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

   2. Simplified 0.2

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 10.7

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]
   5. Using strategy rm

   6. Applied associate-/l* 0.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.45613981804384912 \cdot 10^{278}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -5.31650938160314506 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 3.3409790596422747 \cdot 10^{-221}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 7.8490367773301062 \cdot 10^{81}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))