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

Average Error: 14.6 → 1.4

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \le -1.132598746854114829512768870270445047675 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le -9.566229803325124100269745305371770265409 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le 6.302366347983369041436870740623293996999 \cdot 10^{-316}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \left(y \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le 1.885617898338493029317290072257967706665 \cdot 10^{255}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \left(y \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r4892568 = x;
        double r4892569 = y;
        double r4892570 = z;
        double r4892571 = r4892569 / r4892570;
        double r4892572 = t;
        double r4892573 = r4892571 * r4892572;
        double r4892574 = r4892573 / r4892572;
        double r4892575 = r4892568 * r4892574;
        return r4892575;
}

↓

double f(double x, double y, double z, double t) {
        double r4892576 = y;
        double r4892577 = z;
        double r4892578 = r4892576 / r4892577;
        double r4892579 = t;
        double r4892580 = r4892578 * r4892579;
        double r4892581 = r4892580 / r4892579;
        double r4892582 = -1.1325987468541148e+144;
        bool r4892583 = r4892581 <= r4892582;
        double r4892584 = x;
        double r4892585 = r4892584 / r4892577;
        double r4892586 = r4892585 * r4892576;
        double r4892587 = -9.566229803325124e-252;
        bool r4892588 = r4892581 <= r4892587;
        double r4892589 = r4892584 * r4892581;
        double r4892590 = 6.3023663479834e-316;
        bool r4892591 = r4892581 <= r4892590;
        double r4892592 = cbrt(r4892584);
        double r4892593 = r4892576 * r4892592;
        double r4892594 = r4892592 * r4892593;
        double r4892595 = r4892592 / r4892577;
        double r4892596 = r4892594 * r4892595;
        double r4892597 = 1.885617898338493e+255;
        bool r4892598 = r4892581 <= r4892597;
        double r4892599 = r4892598 ? r4892589 : r4892596;
        double r4892600 = r4892591 ? r4892596 : r4892599;
        double r4892601 = r4892588 ? r4892589 : r4892600;
        double r4892602 = r4892583 ? r4892586 : r4892601;
        return r4892602;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if (/ (* (/ y z) t) t) < -1.1325987468541148e+144

   1. Initial program 37.5

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

   2. Simplified 4.4

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

   if -1.1325987468541148e+144 < (/ (* (/ y z) t) t) < -9.566229803325124e-252 or 6.3023663479834e-316 < (/ (* (/ y z) t) t) < 1.885617898338493e+255

   1. Initial program 0.7

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

   if -9.566229803325124e-252 < (/ (* (/ y z) t) t) < 6.3023663479834e-316 or 1.885617898338493e+255 < (/ (* (/ y z) t) t)

   1. Initial program 32.8

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

   2. Simplified 1.6

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

   4. Applied *-un-lft-identity 1.6

      \[\leadsto y \cdot \frac{x}{\color{blue}{1 \cdot z}}\]

   5. Applied add-cube-cbrt 2.2

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

   6. Applied times-frac 2.2

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

   7. Applied associate-*r* 1.8

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

   8. Simplified 1.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \le -1.132598746854114829512768870270445047675 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le -9.566229803325124100269745305371770265409 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le 6.302366347983369041436870740623293996999 \cdot 10^{-316}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \left(y \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \le 1.885617898338493029317290072257967706665 \cdot 10^{255}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z} \cdot t}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \left(y \cdot \sqrt[3]{x}\right)\right) \cdot \frac{\sqrt[3]{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  (* x (/ (* (/ y z) t) t)))