Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.5 → 8.1

Time: 5.8s

Precision: 64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.36748602107025098 \cdot 10^{-151} \lor \neg \left(a \le 5.37824310183339669 \cdot 10^{-226}\right):\\ \;\;\;\;x + \left(y - \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r523521 = x;
        double r523522 = y;
        double r523523 = r523521 + r523522;
        double r523524 = z;
        double r523525 = t;
        double r523526 = r523524 - r523525;
        double r523527 = r523526 * r523522;
        double r523528 = a;
        double r523529 = r523528 - r523525;
        double r523530 = r523527 / r523529;
        double r523531 = r523523 - r523530;
        return r523531;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r523532 = a;
        double r523533 = -1.367486021070251e-151;
        bool r523534 = r523532 <= r523533;
        double r523535 = 5.378243101833397e-226;
        bool r523536 = r523532 <= r523535;
        double r523537 = !r523536;
        bool r523538 = r523534 || r523537;
        double r523539 = x;
        double r523540 = y;
        double r523541 = z;
        double r523542 = t;
        double r523543 = r523541 - r523542;
        double r523544 = cbrt(r523543);
        double r523545 = r523544 * r523544;
        double r523546 = r523532 - r523542;
        double r523547 = cbrt(r523546);
        double r523548 = r523544 / r523547;
        double r523549 = r523540 / r523547;
        double r523550 = r523548 * r523549;
        double r523551 = r523550 / r523547;
        double r523552 = r523545 * r523551;
        double r523553 = r523540 - r523552;
        double r523554 = r523539 + r523553;
        double r523555 = r523541 * r523540;
        double r523556 = r523555 / r523542;
        double r523557 = r523539 + r523556;
        double r523558 = r523538 ? r523554 : r523557;
        return r523558;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.5
Target	8.7
Herbie	8.1

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -1.367486021070251e-151 or 5.378243101833397e-226 < a

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.4

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

   4. Applied times-frac 9.8

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied times-frac 9.8

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

   8. Applied associate-*l* 9.5

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

   10. Applied associate--l+ 6.9

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

   12. Applied div-inv 7.3

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

   13. Applied associate-*l* 8.1

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

   14. Simplified 7.9

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

   if -1.367486021070251e-151 < a < 5.378243101833397e-226

   1. Initial program 22.1

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

   3. Applied add-cube-cbrt 22.4

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

   4. Applied times-frac 21.0

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

   6. Applied add-cube-cbrt 21.0

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

   7. Applied times-frac 21.0

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

   8. Applied associate-*l* 19.6

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

   10. Applied associate--l+ 12.5

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

   11. Taylor expanded around inf 9.1

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.36748602107025098 \cdot 10^{-151} \lor \neg \left(a \le 5.37824310183339669 \cdot 10^{-226}\right):\\ \;\;\;\;x + \left(y - \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{a - t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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