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

Average Error: 14.5 → 0.3

Time: 10.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.2592621423955549 \cdot 10^{306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.8557707244669514 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.22729521642663493 \cdot 10^{-190}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.2094051295440155 \cdot 10^{253}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r740698 = x;
        double r740699 = y;
        double r740700 = z;
        double r740701 = r740699 / r740700;
        double r740702 = t;
        double r740703 = r740701 * r740702;
        double r740704 = r740703 / r740702;
        double r740705 = r740698 * r740704;
        return r740705;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r740706 = y;
        double r740707 = z;
        double r740708 = r740706 / r740707;
        double r740709 = -2.259262142395555e+306;
        bool r740710 = r740708 <= r740709;
        double r740711 = x;
        double r740712 = r740711 * r740706;
        double r740713 = r740712 / r740707;
        double r740714 = -1.8557707244669514e-271;
        bool r740715 = r740708 <= r740714;
        double r740716 = r740711 * r740708;
        double r740717 = 6.227295216426635e-190;
        bool r740718 = r740708 <= r740717;
        double r740719 = 1.0;
        double r740720 = r740719 / r740707;
        double r740721 = r740712 * r740720;
        double r740722 = 6.209405129544016e+253;
        bool r740723 = r740708 <= r740722;
        double r740724 = r740723 ? r740716 : r740713;
        double r740725 = r740718 ? r740721 : r740724;
        double r740726 = r740715 ? r740716 : r740725;
        double r740727 = r740710 ? r740713 : r740726;
        return r740727;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.5
Target	1.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/ y z) < -2.259262142395555e+306 or 6.209405129544016e+253 < (/ y z)

   1. Initial program 56.0

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

   2. Simplified 49.4

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

   4. Applied add-cube-cbrt 49.7

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity 49.7

      \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac 49.7

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r* 12.5

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 12.5

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]

   9. Taylor expanded around 0 0.4

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

   if -2.259262142395555e+306 < (/ y z) < -1.8557707244669514e-271 or 6.227295216426635e-190 < (/ y z) < 6.209405129544016e+253

   1. Initial program 9.3

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

   2. Simplified 0.2

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

   if -1.8557707244669514e-271 < (/ y z) < 6.227295216426635e-190

   1. Initial program 17.3

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

   2. Simplified 12.1

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

   4. Applied div-inv 12.2

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

   5. Applied associate-*r* 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \le -2.2592621423955549 \cdot 10^{306}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le -1.8557707244669514 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.22729521642663493 \cdot 10^{-190}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{y}{z} \le 6.2094051295440155 \cdot 10^{253}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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)))