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

Average Error: 16.5 → 9.2

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.78725408054180311 \cdot 10^{-108}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\\ \mathbf{elif}\;a \le 1.1307683724034909 \cdot 10^{-188}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\frac{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r691576 = x;
        double r691577 = y;
        double r691578 = r691576 + r691577;
        double r691579 = z;
        double r691580 = t;
        double r691581 = r691579 - r691580;
        double r691582 = r691581 * r691577;
        double r691583 = a;
        double r691584 = r691583 - r691580;
        double r691585 = r691582 / r691584;
        double r691586 = r691578 - r691585;
        return r691586;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r691587 = a;
        double r691588 = -2.787254080541803e-108;
        bool r691589 = r691587 <= r691588;
        double r691590 = x;
        double r691591 = y;
        double r691592 = r691590 + r691591;
        double r691593 = z;
        double r691594 = t;
        double r691595 = r691593 - r691594;
        double r691596 = r691587 - r691594;
        double r691597 = cbrt(r691596);
        double r691598 = r691597 * r691597;
        double r691599 = r691595 / r691598;
        double r691600 = cbrt(r691598);
        double r691601 = r691599 / r691600;
        double r691602 = cbrt(r691591);
        double r691603 = r691602 * r691602;
        double r691604 = cbrt(r691597);
        double r691605 = cbrt(r691604);
        double r691606 = r691605 * r691605;
        double r691607 = r691603 / r691606;
        double r691608 = r691601 * r691607;
        double r691609 = r691602 / r691605;
        double r691610 = r691608 * r691609;
        double r691611 = r691592 - r691610;
        double r691612 = 1.1307683724034909e-188;
        bool r691613 = r691587 <= r691612;
        double r691614 = r691593 * r691591;
        double r691615 = r691614 / r691594;
        double r691616 = r691615 + r691590;
        double r691617 = r691604 * r691604;
        double r691618 = cbrt(r691617);
        double r691619 = r691601 / r691618;
        double r691620 = r691591 / r691605;
        double r691621 = r691619 * r691620;
        double r691622 = r691592 - r691621;
        double r691623 = r691613 ? r691616 : r691622;
        double r691624 = r691589 ? r691611 : r691623;
        return r691624;
}

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	8.5
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if a < -2.787254080541803e-108

   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 8.8

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

   6. Applied add-cube-cbrt 8.9

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

   7. Applied cbrt-prod 8.9

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

   8. Applied *-un-lft-identity 8.9

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

   9. Applied times-frac 8.9

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

   10. Applied associate-*r* 8.8

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

   11. Simplified 8.8

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

   13. Applied add-cube-cbrt 8.9

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

   14. Applied add-cube-cbrt 8.9

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

   15. Applied times-frac 8.9

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

   16. Applied associate-*r* 8.8

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

   if -2.787254080541803e-108 < a < 1.1307683724034909e-188

   1. Initial program 20.0

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

   2. Taylor expanded around inf 9.7

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

   if 1.1307683724034909e-188 < a

   1. Initial program 15.3

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   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 9.2

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

   6. Applied add-cube-cbrt 9.2

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

   7. Applied cbrt-prod 9.2

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

   8. Applied *-un-lft-identity 9.2

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

   9. Applied times-frac 9.2

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

   10. Applied associate-*r* 9.2

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

   11. Simplified 9.2

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

   13. Applied add-cube-cbrt 9.2

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

   14. Applied cbrt-prod 9.2

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

   15. Applied *-un-lft-identity 9.2

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

   16. Applied times-frac 9.2

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

   17. Applied associate-*r* 9.3

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

   18. Simplified 9.3

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.78725408054180311 \cdot 10^{-108}:\\ \;\;\;\;\left(x + y\right) - \left(\frac{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\\ \mathbf{elif}\;a \le 1.1307683724034909 \cdot 10^{-188}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\frac{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))))