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

Average Error: 16.6 → 9.1

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.215811501436430229517507639617501300283 \cdot 10^{-173}:\\ \;\;\;\;\left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) + x\\ \mathbf{elif}\;a \le 8.001727821135775174949972331228956921041 \cdot 10^{-216}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{z - t}} \cdot \left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r29120149 = x;
        double r29120150 = y;
        double r29120151 = r29120149 + r29120150;
        double r29120152 = z;
        double r29120153 = t;
        double r29120154 = r29120152 - r29120153;
        double r29120155 = r29120154 * r29120150;
        double r29120156 = a;
        double r29120157 = r29120156 - r29120153;
        double r29120158 = r29120155 / r29120157;
        double r29120159 = r29120151 - r29120158;
        return r29120159;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r29120160 = a;
        double r29120161 = -1.2158115014364302e-173;
        bool r29120162 = r29120160 <= r29120161;
        double r29120163 = y;
        double r29120164 = z;
        double r29120165 = t;
        double r29120166 = r29120164 - r29120165;
        double r29120167 = r29120160 - r29120165;
        double r29120168 = cbrt(r29120167);
        double r29120169 = r29120168 * r29120168;
        double r29120170 = r29120166 / r29120169;
        double r29120171 = r29120163 / r29120168;
        double r29120172 = r29120170 * r29120171;
        double r29120173 = r29120163 - r29120172;
        double r29120174 = x;
        double r29120175 = r29120173 + r29120174;
        double r29120176 = 8.001727821135775e-216;
        bool r29120177 = r29120160 <= r29120176;
        double r29120178 = r29120163 * r29120164;
        double r29120179 = r29120178 / r29120165;
        double r29120180 = r29120174 + r29120179;
        double r29120181 = cbrt(r29120166);
        double r29120182 = r29120181 * r29120181;
        double r29120183 = r29120182 / r29120168;
        double r29120184 = cbrt(r29120181);
        double r29120185 = r29120184 * r29120184;
        double r29120186 = r29120184 * r29120185;
        double r29120187 = r29120186 / r29120168;
        double r29120188 = r29120187 * r29120171;
        double r29120189 = r29120183 * r29120188;
        double r29120190 = r29120163 - r29120189;
        double r29120191 = r29120190 + r29120174;
        double r29120192 = r29120177 ? r29120180 : r29120191;
        double r29120193 = r29120162 ? r29120175 : r29120192;
        return r29120193;
}

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	8.7
Herbie	9.1

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

Derivation

1. Split input into 3 regimes

2. if a < -1.2158115014364302e-173

   1. Initial program 14.6

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

   3. Applied add-cube-cbrt 14.7

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

   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 associate--l+ 8.8

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

   if -1.2158115014364302e-173 < a < 8.001727821135775e-216

   1. Initial program 21.3

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

   2. Taylor expanded around inf 7.7

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

   if 8.001727821135775e-216 < a

   1. Initial program 16.5

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

   3. Applied add-cube-cbrt 16.6

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

   4. Applied times-frac 10.9

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

   6. Applied associate--l+ 10.4

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

   8. Applied add-cube-cbrt 7.9

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

   9. Applied times-frac 8.0

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

   10. Applied associate-*l* 7.5

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

   12. Applied add-cube-cbrt 10.0

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.215811501436430229517507639617501300283 \cdot 10^{-173}:\\ \;\;\;\;\left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) + x\\ \mathbf{elif}\;a \le 8.001727821135775174949972331228956921041 \cdot 10^{-216}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{z - t}} \cdot \left(\sqrt[3]{\sqrt[3]{z - t}} \cdot \sqrt[3]{\sqrt[3]{z - t}}\right)}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))