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

Average Error: 15.1 → 0.5

Time: 23.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.081925593636316638624250852721330734754 \cdot 10^{268} \lor \neg \left(\frac{y}{z} \le -1.636189466711752142567162663177727587498 \cdot 10^{-114} \lor \neg \left(\frac{y}{z} \le 1.319155274396128272951438676958151064215 \cdot 10^{-321}\right) \land \frac{y}{z} \le 1.104049436697308873782781562368572098325 \cdot 10^{252}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r86540 = x;
        double r86541 = y;
        double r86542 = z;
        double r86543 = r86541 / r86542;
        double r86544 = t;
        double r86545 = r86543 * r86544;
        double r86546 = r86545 / r86544;
        double r86547 = r86540 * r86546;
        return r86547;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r86548 = y;
        double r86549 = z;
        double r86550 = r86548 / r86549;
        double r86551 = -1.0819255936363166e+268;
        bool r86552 = r86550 <= r86551;
        double r86553 = -1.636189466711752e-114;
        bool r86554 = r86550 <= r86553;
        double r86555 = 1.3191552743961e-321;
        bool r86556 = r86550 <= r86555;
        double r86557 = !r86556;
        double r86558 = 1.1040494366973089e+252;
        bool r86559 = r86550 <= r86558;
        bool r86560 = r86557 && r86559;
        bool r86561 = r86554 || r86560;
        double r86562 = !r86561;
        bool r86563 = r86552 || r86562;
        double r86564 = x;
        double r86565 = r86564 / r86549;
        double r86566 = r86548 * r86565;
        double r86567 = r86550 * r86564;
        double r86568 = r86563 ? r86566 : r86567;
        return r86568;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 2 regimes

2. if (/ y z) < -1.0819255936363166e+268 or -1.636189466711752e-114 < (/ y z) < 1.3191552743961e-321 or 1.1040494366973089e+252 < (/ y z)

   1. Initial program 25.2

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

   2. Simplified 17.6

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

   4. Applied div-inv 17.6

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

   5. Applied associate-*l* 1.1

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

   6. Simplified 1.0

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

   if -1.0819255936363166e+268 < (/ y z) < -1.636189466711752e-114 or 1.3191552743961e-321 < (/ y z) < 1.1040494366973089e+252

   1. Initial program 9.8

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

   2. Simplified 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -1.081925593636316638624250852721330734754 \cdot 10^{268} \lor \neg \left(\frac{y}{z} \le -1.636189466711752142567162663177727587498 \cdot 10^{-114} \lor \neg \left(\frac{y}{z} \le 1.319155274396128272951438676958151064215 \cdot 10^{-321}\right) \land \frac{y}{z} \le 1.104049436697308873782781562368572098325 \cdot 10^{252}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))