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

Average Error: 16.3 → 8.6

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.329630304780208437947829967576919046 \cdot 10^{-160} \lor \neg \left(a \le 4.776220601035187966235421470923183704471 \cdot 10^{-100}\right):\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r485455 = x;
        double r485456 = y;
        double r485457 = r485455 + r485456;
        double r485458 = z;
        double r485459 = t;
        double r485460 = r485458 - r485459;
        double r485461 = r485460 * r485456;
        double r485462 = a;
        double r485463 = r485462 - r485459;
        double r485464 = r485461 / r485463;
        double r485465 = r485457 - r485464;
        return r485465;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r485466 = a;
        double r485467 = -7.329630304780208e-160;
        bool r485468 = r485466 <= r485467;
        double r485469 = 4.776220601035188e-100;
        bool r485470 = r485466 <= r485469;
        double r485471 = !r485470;
        bool r485472 = r485468 || r485471;
        double r485473 = x;
        double r485474 = y;
        double r485475 = cbrt(r485474);
        double r485476 = r485475 * r485475;
        double r485477 = t;
        double r485478 = r485466 - r485477;
        double r485479 = cbrt(r485478);
        double r485480 = r485479 * r485479;
        double r485481 = r485476 / r485480;
        double r485482 = r485475 / r485479;
        double r485483 = z;
        double r485484 = r485483 - r485477;
        double r485485 = r485482 * r485484;
        double r485486 = r485481 * r485485;
        double r485487 = r485474 - r485486;
        double r485488 = r485473 + r485487;
        double r485489 = r485474 * r485483;
        double r485490 = r485489 / r485477;
        double r485491 = r485473 + r485490;
        double r485492 = r485472 ? r485488 : r485491;
        return r485492;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.3
Target	8.4
Herbie	8.6

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

Derivation

1. Split input into 2 regimes

2. if a < -7.329630304780208e-160 or 4.776220601035188e-100 < a

   1. Initial program 14.7

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

   3. Applied associate--l+ 13.1

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

   4. Simplified 7.3

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

   6. Applied add-cube-cbrt 8.9

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

   7. Applied add-cube-cbrt 9.1

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

   8. Applied times-frac 9.2

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

   9. Applied associate-*l* 8.1

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

   10. Simplified 8.1

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

   if -7.329630304780208e-160 < a < 4.776220601035188e-100

   1. Initial program 20.8

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

   2. Taylor expanded around inf 10.1

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

   3. Simplified 10.1

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.329630304780208437947829967576919046 \cdot 10^{-160} \lor \neg \left(a \le 4.776220601035187966235421470923183704471 \cdot 10^{-100}\right):\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(z - t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))))