Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Average Error: 1.5 → 1.0

Time: 27.5s

Precision: 64

Math

\[x + y \cdot \frac{z - t}{z - a}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.212645841190267324612631053459760197997 \cdot 10^{-233}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \le 1.119067119670475321537633852374314219182 \cdot 10^{-184}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r176586142 = x;
        double r176586143 = y;
        double r176586144 = z;
        double r176586145 = t;
        double r176586146 = r176586144 - r176586145;
        double r176586147 = a;
        double r176586148 = r176586144 - r176586147;
        double r176586149 = r176586146 / r176586148;
        double r176586150 = r176586143 * r176586149;
        double r176586151 = r176586142 + r176586150;
        return r176586151;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r176586152 = y;
        double r176586153 = -2.2126458411902673e-233;
        bool r176586154 = r176586152 <= r176586153;
        double r176586155 = x;
        double r176586156 = z;
        double r176586157 = a;
        double r176586158 = r176586156 - r176586157;
        double r176586159 = t;
        double r176586160 = r176586156 - r176586159;
        double r176586161 = r176586158 / r176586160;
        double r176586162 = r176586152 / r176586161;
        double r176586163 = r176586155 + r176586162;
        double r176586164 = 1.1190671196704753e-184;
        bool r176586165 = r176586152 <= r176586164;
        double r176586166 = r176586160 * r176586152;
        double r176586167 = r176586166 / r176586158;
        double r176586168 = r176586155 + r176586167;
        double r176586169 = r176586165 ? r176586168 : r176586163;
        double r176586170 = r176586154 ? r176586163 : r176586169;
        return r176586170;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.5
Target	1.4
Herbie	1.0

\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

1. Split input into 2 regimes

2. if y < -2.2126458411902673e-233 or 1.1190671196704753e-184 < y

   1. Initial program 1.2

      \[x + y \cdot \frac{z - t}{z - a}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.8

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

   4. Applied associate-*l* 1.8

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \frac{z - t}{z - a}\right)}\]
   5. Using strategy rm

   6. Applied pow1 1.8

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

   7. Applied pow1 1.8

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

   8. Applied pow-prod-down 1.8

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

   9. Applied pow1 1.8

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

   10. Applied pow1 1.8

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

   11. Applied pow-prod-down 1.8

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

   12. Applied pow-prod-down 1.8

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

   13. Simplified 1.1

       \[\leadsto x + {\color{blue}{\left(\frac{y}{\frac{z - a}{z - t}}\right)}}^{1}\]

   if -2.2126458411902673e-233 < y < 1.1190671196704753e-184

   1. Initial program 3.0

      \[x + y \cdot \frac{z - t}{z - a}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 3.1

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

   4. Applied associate-*l* 3.1

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{y} \cdot \frac{z - t}{z - a}\right)}\]
   5. Using strategy rm

   6. Applied associate-*r/ 1.3

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

   7. Applied associate-*r/ 0.6

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

   8. Simplified 0.4

      \[\leadsto x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{z - a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.212645841190267324612631053459760197997 \cdot 10^{-233}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{elif}\;y \le 1.119067119670475321537633852374314219182 \cdot 10^{-184}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (* y (/ (- z t) (- z a)))))