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

Average Error: 16.3 → 8.9

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.177760378531967537535362926649167713566 \cdot 10^{-164}:\\ \;\;\;\;\left(x + y\right) - \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{z - t}{\left(\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right)}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 5.253807105661921599150117490131368798832 \cdot 10^{-287}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{z - t}{\left(\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r30386115 = x;
        double r30386116 = y;
        double r30386117 = r30386115 + r30386116;
        double r30386118 = z;
        double r30386119 = t;
        double r30386120 = r30386118 - r30386119;
        double r30386121 = r30386120 * r30386116;
        double r30386122 = a;
        double r30386123 = r30386122 - r30386119;
        double r30386124 = r30386121 / r30386123;
        double r30386125 = r30386117 - r30386124;
        return r30386125;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r30386126 = x;
        double r30386127 = y;
        double r30386128 = r30386126 + r30386127;
        double r30386129 = z;
        double r30386130 = t;
        double r30386131 = r30386129 - r30386130;
        double r30386132 = r30386131 * r30386127;
        double r30386133 = a;
        double r30386134 = r30386133 - r30386130;
        double r30386135 = r30386132 / r30386134;
        double r30386136 = r30386128 - r30386135;
        double r30386137 = -3.1777603785319675e-164;
        bool r30386138 = r30386136 <= r30386137;
        double r30386139 = cbrt(r30386134);
        double r30386140 = cbrt(r30386139);
        double r30386141 = cbrt(r30386140);
        double r30386142 = r30386127 / r30386141;
        double r30386143 = r30386139 * r30386139;
        double r30386144 = cbrt(r30386143);
        double r30386145 = r30386143 * r30386144;
        double r30386146 = r30386141 * r30386141;
        double r30386147 = r30386145 * r30386146;
        double r30386148 = r30386131 / r30386147;
        double r30386149 = r30386142 * r30386148;
        double r30386150 = r30386128 - r30386149;
        double r30386151 = 5.253807105661922e-287;
        bool r30386152 = r30386136 <= r30386151;
        double r30386153 = r30386129 * r30386127;
        double r30386154 = r30386153 / r30386130;
        double r30386155 = r30386154 + r30386126;
        double r30386156 = r30386152 ? r30386155 : r30386150;
        double r30386157 = r30386138 ? r30386150 : r30386156;
        return r30386157;
}

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	8.3
Herbie	8.9

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

Derivation

1. Split input into 2 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -3.1777603785319675e-164 or 5.253807105661922e-287 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 12.6

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

   3. Applied add-cube-cbrt 12.8

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

   4. Applied times-frac 7.3

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

   6. Applied add-cube-cbrt 7.3

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

   7. Applied cbrt-prod 7.3

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

   8. Applied *-un-lft-identity 7.3

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

   9. Applied times-frac 7.3

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

   10. Applied associate-*r* 7.4

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

   11. Simplified 7.4

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

   13. Applied add-cube-cbrt 7.5

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

   14. Applied *-un-lft-identity 7.5

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

   15. Applied times-frac 7.5

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

   16. Applied associate-*r* 7.6

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

   17. Simplified 7.6

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

   if -3.1777603785319675e-164 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 5.253807105661922e-287

   1. Initial program 49.8

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

   2. Taylor expanded around inf 20.4

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -3.177760378531967537535362926649167713566 \cdot 10^{-164}:\\ \;\;\;\;\left(x + y\right) - \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{z - t}{\left(\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right)}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 5.253807105661921599150117490131368798832 \cdot 10^{-287}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{z - t}{\left(\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))