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

Average Error: 16.0 → 10.1

Time: 25.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.3731458525563276 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.9186525251190146 \cdot 10^{+129}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r30751431 = x;
        double r30751432 = y;
        double r30751433 = r30751431 + r30751432;
        double r30751434 = z;
        double r30751435 = t;
        double r30751436 = r30751434 - r30751435;
        double r30751437 = r30751436 * r30751432;
        double r30751438 = a;
        double r30751439 = r30751438 - r30751435;
        double r30751440 = r30751437 / r30751439;
        double r30751441 = r30751433 - r30751440;
        return r30751441;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r30751442 = t;
        double r30751443 = -1.3731458525563276e+50;
        bool r30751444 = r30751442 <= r30751443;
        double r30751445 = x;
        double r30751446 = z;
        double r30751447 = y;
        double r30751448 = r30751446 * r30751447;
        double r30751449 = r30751448 / r30751442;
        double r30751450 = r30751445 + r30751449;
        double r30751451 = 2.9186525251190146e+129;
        bool r30751452 = r30751442 <= r30751451;
        double r30751453 = r30751445 + r30751447;
        double r30751454 = r30751446 - r30751442;
        double r30751455 = a;
        double r30751456 = r30751455 - r30751442;
        double r30751457 = cbrt(r30751456);
        double r30751458 = cbrt(r30751457);
        double r30751459 = r30751447 / r30751458;
        double r30751460 = cbrt(r30751459);
        double r30751461 = 1.0;
        double r30751462 = r30751458 * r30751458;
        double r30751463 = r30751461 / r30751462;
        double r30751464 = cbrt(r30751463);
        double r30751465 = r30751460 * r30751464;
        double r30751466 = r30751457 / r30751465;
        double r30751467 = r30751447 / r30751457;
        double r30751468 = cbrt(r30751467);
        double r30751469 = r30751457 / r30751468;
        double r30751470 = r30751466 * r30751469;
        double r30751471 = r30751454 / r30751470;
        double r30751472 = r30751471 * r30751468;
        double r30751473 = r30751453 - r30751472;
        double r30751474 = r30751452 ? r30751473 : r30751450;
        double r30751475 = r30751444 ? r30751450 : r30751474;
        return r30751475;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.0
Target	8.2
Herbie	10.1

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.3731458525563276e+50 or 2.9186525251190146e+129 < t

   1. Initial program 28.8

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

   2. Taylor expanded around inf 17.7

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

   if -1.3731458525563276e+50 < t < 2.9186525251190146e+129

   1. Initial program 8.1

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

   3. Applied add-cube-cbrt 8.3

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

   4. Applied times-frac 5.9

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

   6. Applied add-cube-cbrt 6.0

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

   7. Applied associate-*r* 5.9

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

   8. Simplified 5.4

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

   10. Applied add-cube-cbrt 5.4

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

   11. Applied *-un-lft-identity 5.4

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

   12. Applied times-frac 5.4

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

   13. Applied cbrt-prod 5.4

       \[\leadsto \left(x + y\right) - \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}} \cdot \frac{\sqrt[3]{a - t}}{\color{blue}{\sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3731458525563276 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.9186525251190146 \cdot 10^{+129}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :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))))