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

Average Error: 16.1 → 6.6

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -9.203641485734956442447032350818823636586 \cdot 10^{-277}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r340162 = x;
        double r340163 = y;
        double r340164 = r340162 + r340163;
        double r340165 = z;
        double r340166 = t;
        double r340167 = r340165 - r340166;
        double r340168 = r340167 * r340163;
        double r340169 = a;
        double r340170 = r340169 - r340166;
        double r340171 = r340168 / r340170;
        double r340172 = r340164 - r340171;
        return r340172;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r340173 = x;
        double r340174 = y;
        double r340175 = r340173 + r340174;
        double r340176 = z;
        double r340177 = t;
        double r340178 = r340176 - r340177;
        double r340179 = r340178 * r340174;
        double r340180 = a;
        double r340181 = r340180 - r340177;
        double r340182 = r340179 / r340181;
        double r340183 = r340175 - r340182;
        double r340184 = -inf.0;
        bool r340185 = r340183 <= r340184;
        double r340186 = r340177 / r340181;
        double r340187 = r340176 / r340181;
        double r340188 = r340186 - r340187;
        double r340189 = fma(r340174, r340188, r340174);
        double r340190 = r340189 + r340173;
        double r340191 = cbrt(r340190);
        double r340192 = r340191 * r340191;
        double r340193 = cbrt(r340191);
        double r340194 = r340193 * r340193;
        double r340195 = r340194 * r340193;
        double r340196 = r340192 * r340195;
        double r340197 = -9.203641485734956e-277;
        bool r340198 = r340183 <= r340197;
        double r340199 = 0.0;
        bool r340200 = r340183 <= r340199;
        double r340201 = r340176 / r340177;
        double r340202 = fma(r340201, r340174, r340173);
        double r340203 = r340200 ? r340202 : r340196;
        double r340204 = r340198 ? r340183 : r340203;
        double r340205 = r340185 ? r340196 : r340204;
        return r340205;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.1
Target	8.1
Herbie	6.6

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

Derivation

1. Split input into 3 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -inf.0 or 0.0 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 19.9

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

   2. Simplified 10.0

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

   4. Applied div-sub 9.9

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

   6. Applied add-cube-cbrt 10.1

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

   7. Applied *-un-lft-identity 10.1

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

   8. Applied add-cube-cbrt 10.2

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

   9. Applied times-frac 10.2

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

   10. Applied prod-diff 10.2

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

   11. Simplified 10.0

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

   12. Simplified 10.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{t} \cdot \sqrt[3]{t}, \frac{\sqrt[3]{t}}{a - t}, \frac{-z}{a - t}\right) + \color{blue}{0}, y, x + y\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 11.0

       \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{t} \cdot \sqrt[3]{t}, \frac{\sqrt[3]{t}}{a - t}, \frac{-z}{a - t}\right) + 0, y, x + y\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{t} \cdot \sqrt[3]{t}, \frac{\sqrt[3]{t}}{a - t}, \frac{-z}{a - t}\right) + 0, y, x + y\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{t} \cdot \sqrt[3]{t}, \frac{\sqrt[3]{t}}{a - t}, \frac{-z}{a - t}\right) + 0, y, x + y\right)}}\]

   15. Simplified 11.0

       \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{t} \cdot \sqrt[3]{t}, \frac{\sqrt[3]{t}}{a - t}, \frac{-z}{a - t}\right) + 0, y, x + y\right)}\]

   16. Simplified 8.2

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

   18. Applied add-cube-cbrt 8.5

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

   if -inf.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < -9.203641485734956e-277

   1. Initial program 1.3

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

   if -9.203641485734956e-277 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

   1. Initial program 60.5

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

   2. Simplified 60.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{a - t}, y, x + y\right)}\]

   3. Taylor expanded around inf 18.5

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

   4. Simplified 18.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -9.203641485734956442447032350818823636586 \cdot 10^{-277}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(y, \frac{t}{a - t} - \frac{z}{a - t}, y\right) + x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))))