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

Average Error: 16.9 → 9.1

Time: 28.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -9.4689010464121415508002148715762489261 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a - t}, y, x + y\right)\\ \mathbf{elif}\;a \le 1.019767468767238742050870882472042955611 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}} + \left(x + y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r372253 = x;
        double r372254 = y;
        double r372255 = r372253 + r372254;
        double r372256 = z;
        double r372257 = t;
        double r372258 = r372256 - r372257;
        double r372259 = r372258 * r372254;
        double r372260 = a;
        double r372261 = r372260 - r372257;
        double r372262 = r372259 / r372261;
        double r372263 = r372255 - r372262;
        return r372263;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r372264 = a;
        double r372265 = -9.468901046412142e-153;
        bool r372266 = r372264 <= r372265;
        double r372267 = t;
        double r372268 = z;
        double r372269 = r372267 - r372268;
        double r372270 = 1.0;
        double r372271 = r372264 - r372267;
        double r372272 = r372270 / r372271;
        double r372273 = r372269 * r372272;
        double r372274 = y;
        double r372275 = x;
        double r372276 = r372275 + r372274;
        double r372277 = fma(r372273, r372274, r372276);
        double r372278 = 1.0197674687672387e-151;
        bool r372279 = r372264 <= r372278;
        double r372280 = r372268 / r372267;
        double r372281 = fma(r372280, r372274, r372275);
        double r372282 = cbrt(r372271);
        double r372283 = r372282 * r372282;
        double r372284 = r372274 / r372283;
        double r372285 = r372269 / r372282;
        double r372286 = r372284 * r372285;
        double r372287 = r372286 + r372276;
        double r372288 = r372279 ? r372281 : r372287;
        double r372289 = r372266 ? r372277 : r372288;
        return r372289;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.9
Target	8.5
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 < -9.468901046412142e-153

   1. Initial program 15.4

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

   2. Simplified 8.5

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

   4. Applied div-inv 8.5

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

   if -9.468901046412142e-153 < a < 1.0197674687672387e-151

   1. Initial program 21.6

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

   2. Simplified 21.1

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

   3. Taylor expanded around inf 9.2

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

   4. Simplified 9.1

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

   if 1.0197674687672387e-151 < a

   1. Initial program 15.6

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

   2. Simplified 9.2

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

   4. Applied div-inv 9.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - z\right) \cdot \frac{1}{a - t}}, y, x + y\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 9.3

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

   7. Applied add-cube-cbrt 9.3

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

   8. Applied times-frac 9.3

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

   9. Simplified 9.3

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

   10. Simplified 9.3

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

   12. Applied fma-udef 9.3

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

   13. Simplified 9.2

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

   15. Applied add-cube-cbrt 9.3

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

   16. Applied *-un-lft-identity 9.3

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

   17. Applied times-frac 9.3

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

   18. Applied associate-*r* 9.7

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

   19. Simplified 9.7

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.4689010464121415508002148715762489261 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{1}{a - t}, y, x + y\right)\\ \mathbf{elif}\;a \le 1.019767468767238742050870882472042955611 \cdot 10^{-151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{t - z}{\sqrt[3]{a - t}} + \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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-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))))