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

Average Error: 16.4 → 10.3

Time: 54.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.37314585255632761189109605897833700965 \cdot 10^{50}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.918652525119014591089469715053014371125 \cdot 10^{129}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r30986596 = x;
        double r30986597 = y;
        double r30986598 = r30986596 + r30986597;
        double r30986599 = z;
        double r30986600 = t;
        double r30986601 = r30986599 - r30986600;
        double r30986602 = r30986601 * r30986597;
        double r30986603 = a;
        double r30986604 = r30986603 - r30986600;
        double r30986605 = r30986602 / r30986604;
        double r30986606 = r30986598 - r30986605;
        return r30986606;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r30986607 = t;
        double r30986608 = -1.3731458525563276e+50;
        bool r30986609 = r30986607 <= r30986608;
        double r30986610 = x;
        double r30986611 = z;
        double r30986612 = y;
        double r30986613 = r30986611 * r30986612;
        double r30986614 = r30986613 / r30986607;
        double r30986615 = r30986610 + r30986614;
        double r30986616 = 2.9186525251190146e+129;
        bool r30986617 = r30986607 <= r30986616;
        double r30986618 = r30986610 + r30986612;
        double r30986619 = r30986611 - r30986607;
        double r30986620 = a;
        double r30986621 = r30986620 - r30986607;
        double r30986622 = cbrt(r30986621);
        double r30986623 = cbrt(r30986622);
        double r30986624 = r30986612 / r30986623;
        double r30986625 = cbrt(r30986624);
        double r30986626 = 1.0;
        double r30986627 = r30986623 * r30986623;
        double r30986628 = r30986626 / r30986627;
        double r30986629 = cbrt(r30986628);
        double r30986630 = r30986625 * r30986629;
        double r30986631 = r30986622 / r30986630;
        double r30986632 = r30986612 / r30986622;
        double r30986633 = cbrt(r30986632);
        double r30986634 = r30986622 / r30986633;
        double r30986635 = r30986631 * r30986634;
        double r30986636 = r30986619 / r30986635;
        double r30986637 = r30986636 * r30986633;
        double r30986638 = r30986618 - r30986637;
        double r30986639 = r30986617 ? r30986638 : r30986615;
        double r30986640 = r30986609 ? r30986615 : r30986639;
        return r30986640;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.4
Target	8.3
Herbie	10.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 29.4

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

   2. Taylor expanded around inf 18.1

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

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

   1. Initial program 8.4

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

   3. Applied add-cube-cbrt 8.6

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

   4. Applied times-frac 6.0

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

   6. Applied add-cube-cbrt 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* 6.0

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

   8. Simplified 5.5

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\]
   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 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.37314585255632761189109605897833700965 \cdot 10^{50}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.918652525119014591089469715053014371125 \cdot 10^{129}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array}\]

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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

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