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

Average Error: 16.3 → 9.2

Time: 25.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.838369765495349463260835321405014535092 \cdot 10^{-148}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 3.265578407157981999998792060881541473299 \cdot 10^{-103}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r32664532 = x;
        double r32664533 = y;
        double r32664534 = r32664532 + r32664533;
        double r32664535 = z;
        double r32664536 = t;
        double r32664537 = r32664535 - r32664536;
        double r32664538 = r32664537 * r32664533;
        double r32664539 = a;
        double r32664540 = r32664539 - r32664536;
        double r32664541 = r32664538 / r32664540;
        double r32664542 = r32664534 - r32664541;
        return r32664542;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r32664543 = a;
        double r32664544 = -1.8383697654953495e-148;
        bool r32664545 = r32664543 <= r32664544;
        double r32664546 = x;
        double r32664547 = y;
        double r32664548 = r32664546 + r32664547;
        double r32664549 = z;
        double r32664550 = t;
        double r32664551 = r32664549 - r32664550;
        double r32664552 = r32664543 - r32664550;
        double r32664553 = cbrt(r32664552);
        double r32664554 = r32664553 * r32664553;
        double r32664555 = cbrt(r32664554);
        double r32664556 = r32664554 * r32664555;
        double r32664557 = r32664551 / r32664556;
        double r32664558 = cbrt(r32664553);
        double r32664559 = r32664547 / r32664558;
        double r32664560 = r32664557 * r32664559;
        double r32664561 = r32664548 - r32664560;
        double r32664562 = 3.265578407157982e-103;
        bool r32664563 = r32664543 <= r32664562;
        double r32664564 = r32664549 * r32664547;
        double r32664565 = r32664564 / r32664550;
        double r32664566 = r32664546 + r32664565;
        double r32664567 = cbrt(r32664551);
        double r32664568 = r32664553 * r32664558;
        double r32664569 = r32664567 / r32664568;
        double r32664570 = r32664569 * r32664559;
        double r32664571 = r32664567 * r32664567;
        double r32664572 = r32664571 / r32664568;
        double r32664573 = r32664570 * r32664572;
        double r32664574 = r32664548 - r32664573;
        double r32664575 = r32664563 ? r32664566 : r32664574;
        double r32664576 = r32664545 ? r32664561 : r32664575;
        return r32664576;
}

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	8.7
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if a < -1.8383697654953495e-148

   1. Initial program 15.5

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

   3. Applied add-cube-cbrt 15.6

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

   4. Applied times-frac 9.0

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied cbrt-prod 9.1

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

   8. Applied *-un-lft-identity 9.1

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

   9. Applied times-frac 9.1

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

   10. Applied associate-*r* 8.9

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

   11. Simplified 8.9

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

   if -1.8383697654953495e-148 < a < 3.265578407157982e-103

   1. Initial program 19.9

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

   2. Taylor expanded around inf 10.7

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

   if 3.265578407157982e-103 < a

   1. Initial program 14.4

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

   3. Applied add-cube-cbrt 14.6

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

   4. Applied times-frac 8.6

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

   6. Applied add-cube-cbrt 8.6

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

   7. Applied *-un-lft-identity 8.6

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

   8. Applied times-frac 8.6

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

   9. Applied associate-*r* 8.5

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

   10. Simplified 8.5

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

   12. Applied add-cube-cbrt 8.5

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

   13. Applied times-frac 8.5

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

   14. Applied associate-*l* 8.2

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.838369765495349463260835321405014535092 \cdot 10^{-148}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 3.265578407157981999998792060881541473299 \cdot 10^{-103}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{\sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))