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

Average Error: 15.1 → 2.7

Time: 13.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.311589734371419027279238960116033835921 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 2.460446916289407789999312588483742434378 \cdot 10^{-321}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le 2.676343230069060529690221971470663563109 \cdot 10^{68}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4539932 = x;
        double r4539933 = y;
        double r4539934 = z;
        double r4539935 = r4539933 / r4539934;
        double r4539936 = t;
        double r4539937 = r4539935 * r4539936;
        double r4539938 = r4539937 / r4539936;
        double r4539939 = r4539932 * r4539938;
        return r4539939;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r4539940 = y;
        double r4539941 = z;
        double r4539942 = r4539940 / r4539941;
        double r4539943 = -2.311589734371419e-251;
        bool r4539944 = r4539942 <= r4539943;
        double r4539945 = x;
        double r4539946 = r4539941 / r4539940;
        double r4539947 = r4539945 / r4539946;
        double r4539948 = 2.4604469162894e-321;
        bool r4539949 = r4539942 <= r4539948;
        double r4539950 = r4539945 / r4539941;
        double r4539951 = r4539950 * r4539940;
        double r4539952 = 2.6763432300690605e+68;
        bool r4539953 = r4539942 <= r4539952;
        double r4539954 = r4539953 ? r4539947 : r4539951;
        double r4539955 = r4539949 ? r4539951 : r4539954;
        double r4539956 = r4539944 ? r4539947 : r4539955;
        return r4539956;
}

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) < -2.311589734371419e-251 or 2.4604469162894e-321 < (/ y z) < 2.6763432300690605e+68

   1. Initial program 12.0

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

   2. Simplified 8.0

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

   4. Applied associate-/l* 2.8

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

   if -2.311589734371419e-251 < (/ y z) < 2.4604469162894e-321 or 2.6763432300690605e+68 < (/ y z)

   1. Initial program 21.9

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

   2. Simplified 2.4

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

   4. Applied associate-/l* 13.6

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

   6. Applied associate-/r/ 2.5

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.311589734371419027279238960116033835921 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{y}{z} \le 2.460446916289407789999312588483742434378 \cdot 10^{-321}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{y}{z} \le 2.676343230069060529690221971470663563109 \cdot 10^{68}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))