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

Average Error: 16.6 → 9.7

Time: 27.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -9.69352787633815922929071472670459335417 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 1.889084039169875659727922857236046312161 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 5.919178498880237805604501360892005847147 \cdot 10^{-44}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 9.934875400147095918607310666158699856074 \cdot 10^{-12}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r398800 = x;
        double r398801 = y;
        double r398802 = r398800 + r398801;
        double r398803 = z;
        double r398804 = t;
        double r398805 = r398803 - r398804;
        double r398806 = r398805 * r398801;
        double r398807 = a;
        double r398808 = r398807 - r398804;
        double r398809 = r398806 / r398808;
        double r398810 = r398802 - r398809;
        return r398810;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r398811 = a;
        double r398812 = -9.693527876338159e-114;
        bool r398813 = r398811 <= r398812;
        double r398814 = x;
        double r398815 = y;
        double r398816 = r398814 + r398815;
        double r398817 = z;
        double r398818 = t;
        double r398819 = r398817 - r398818;
        double r398820 = cbrt(r398819);
        double r398821 = r398811 - r398818;
        double r398822 = cbrt(r398821);
        double r398823 = r398820 / r398822;
        double r398824 = r398822 * r398822;
        double r398825 = cbrt(r398824);
        double r398826 = r398823 / r398825;
        double r398827 = cbrt(r398822);
        double r398828 = r398815 / r398827;
        double r398829 = r398826 * r398828;
        double r398830 = r398820 * r398820;
        double r398831 = r398830 / r398822;
        double r398832 = r398829 * r398831;
        double r398833 = cbrt(r398832);
        double r398834 = r398833 * r398833;
        double r398835 = r398834 * r398833;
        double r398836 = r398816 - r398835;
        double r398837 = 1.8890840391698757e-110;
        bool r398838 = r398811 <= r398837;
        double r398839 = r398817 * r398815;
        double r398840 = r398839 / r398818;
        double r398841 = r398840 + r398814;
        double r398842 = 5.919178498880238e-44;
        bool r398843 = r398811 <= r398842;
        double r398844 = r398819 / r398824;
        double r398845 = cbrt(r398844);
        double r398846 = r398845 * r398845;
        double r398847 = r398815 / r398822;
        double r398848 = r398845 * r398847;
        double r398849 = r398846 * r398848;
        double r398850 = r398816 - r398849;
        double r398851 = 9.934875400147096e-12;
        bool r398852 = r398811 <= r398851;
        double r398853 = r398821 / r398815;
        double r398854 = r398819 / r398853;
        double r398855 = r398816 - r398854;
        double r398856 = r398852 ? r398841 : r398855;
        double r398857 = r398843 ? r398850 : r398856;
        double r398858 = r398838 ? r398841 : r398857;
        double r398859 = r398813 ? r398836 : r398858;
        return r398859;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.6
Target	8.4
Herbie	9.7

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

Derivation

1. Split input into 4 regimes

2. if a < -9.693527876338159e-114

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.3

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

   4. Applied times-frac 8.4

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

   6. Applied add-cube-cbrt 8.4

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

   7. Applied cbrt-prod 8.5

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

   8. Applied *-un-lft-identity 8.5

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

   9. Applied times-frac 8.5

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

   10. Applied associate-*r* 8.5

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

   11. Simplified 8.4

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

   13. Applied *-un-lft-identity 8.4

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

   14. Applied add-cube-cbrt 8.5

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

   15. Applied times-frac 8.5

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

   16. Applied times-frac 8.5

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

   17. Applied associate-*l* 8.3

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

   19. Applied add-cube-cbrt 8.3

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

   20. Simplified 8.3

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

   21. Simplified 8.3

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

   if -9.693527876338159e-114 < a < 1.8890840391698757e-110 or 5.919178498880238e-44 < a < 9.934875400147096e-12

   1. Initial program 20.0

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

   2. Taylor expanded around inf 12.5

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

   if 1.8890840391698757e-110 < a < 5.919178498880238e-44

   1. Initial program 15.6

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

   3. Applied add-cube-cbrt 15.9

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

   4. Applied times-frac 14.9

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

   6. Applied add-cube-cbrt 14.9

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

   7. Applied associate-*l* 14.9

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

   if 9.934875400147096e-12 < a

   1. Initial program 14.3

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

   3. Applied associate-/l* 6.8

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.69352787633815922929071472670459335417 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 1.889084039169875659727922857236046312161 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 5.919178498880237805604501360892005847147 \cdot 10^{-44}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 9.934875400147095918607310666158699856074 \cdot 10^{-12}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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-7) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.47542934445772333e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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