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

Average Error: 16.5 → 10.1

Time: 17.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -8.77698945948147 \cdot 10^{-231}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 5.06495798850060336 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r291994 = x;
        double r291995 = y;
        double r291996 = r291994 + r291995;
        double r291997 = z;
        double r291998 = t;
        double r291999 = r291997 - r291998;
        double r292000 = r291999 * r291995;
        double r292001 = a;
        double r292002 = r292001 - r291998;
        double r292003 = r292000 / r292002;
        double r292004 = r291996 - r292003;
        return r292004;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r292005 = a;
        double r292006 = -8.77698945948147e-231;
        bool r292007 = r292005 <= r292006;
        double r292008 = x;
        double r292009 = y;
        double r292010 = r292008 + r292009;
        double r292011 = z;
        double r292012 = t;
        double r292013 = r292011 - r292012;
        double r292014 = cbrt(r292013);
        double r292015 = 1.0;
        double r292016 = r292005 - r292012;
        double r292017 = cbrt(r292016);
        double r292018 = r292017 * r292017;
        double r292019 = r292015 / r292018;
        double r292020 = cbrt(r292019);
        double r292021 = r292014 * r292020;
        double r292022 = r292013 / r292018;
        double r292023 = cbrt(r292022);
        double r292024 = r292021 * r292023;
        double r292025 = r292009 / r292017;
        double r292026 = r292014 * r292025;
        double r292027 = cbrt(r292018);
        double r292028 = r292026 / r292027;
        double r292029 = r292024 * r292028;
        double r292030 = r292010 - r292029;
        double r292031 = 5.0649579885006034e-82;
        bool r292032 = r292005 <= r292031;
        double r292033 = r292011 * r292009;
        double r292034 = r292033 / r292012;
        double r292035 = r292034 + r292008;
        double r292036 = r292014 * r292014;
        double r292037 = r292036 / r292017;
        double r292038 = cbrt(r292037);
        double r292039 = r292014 / r292017;
        double r292040 = cbrt(r292039);
        double r292041 = r292038 * r292040;
        double r292042 = r292041 * r292025;
        double r292043 = r292024 * r292042;
        double r292044 = r292010 - r292043;
        double r292045 = r292032 ? r292035 : r292044;
        double r292046 = r292007 ? r292030 : r292045;
        return r292046;
}

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	8.6
Herbie	10.1

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

Derivation

1. Split input into 3 regimes

2. if a < -8.77698945948147e-231

   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 10.2

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

   6. Applied add-cube-cbrt 10.3

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

   7. Applied associate-*l* 10.3

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

   9. Applied div-inv 10.3

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

   10. Applied cbrt-prod 10.3

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

   12. Applied cbrt-div 10.3

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

   13. Applied associate-*l/ 10.3

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

   if -8.77698945948147e-231 < a < 5.0649579885006034e-82

   1. Initial program 20.2

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

   2. Taylor expanded around inf 11.2

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

   if 5.0649579885006034e-82 < a

   1. Initial program 15.8

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   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 9.1

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied associate-*l* 9.1

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

   9. Applied div-inv 9.1

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

   10. Applied cbrt-prod 9.1

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

   12. Applied add-cube-cbrt 9.1

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

   13. Applied times-frac 9.1

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

   14. Applied cbrt-prod 9.1

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.77698945948147 \cdot 10^{-231}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \frac{y}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 5.06495798850060336 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))