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

Average Error: 16.0 → 6.9

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.065617326870013528561467193287073133332 \cdot 10^{-277}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r400372 = x;
        double r400373 = y;
        double r400374 = r400372 + r400373;
        double r400375 = z;
        double r400376 = t;
        double r400377 = r400375 - r400376;
        double r400378 = r400377 * r400373;
        double r400379 = a;
        double r400380 = r400379 - r400376;
        double r400381 = r400378 / r400380;
        double r400382 = r400374 - r400381;
        return r400382;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r400383 = x;
        double r400384 = y;
        double r400385 = r400383 + r400384;
        double r400386 = z;
        double r400387 = t;
        double r400388 = r400386 - r400387;
        double r400389 = r400388 * r400384;
        double r400390 = a;
        double r400391 = r400390 - r400387;
        double r400392 = r400389 / r400391;
        double r400393 = r400385 - r400392;
        double r400394 = -inf.0;
        bool r400395 = r400393 <= r400394;
        double r400396 = r400386 / r400387;
        double r400397 = fma(r400396, r400384, r400383);
        double r400398 = -1.0656173268700135e-277;
        bool r400399 = r400393 <= r400398;
        double r400400 = 0.0;
        bool r400401 = r400393 <= r400400;
        double r400402 = r400386 * r400384;
        double r400403 = r400402 / r400387;
        double r400404 = r400403 + r400383;
        double r400405 = r400387 - r400386;
        double r400406 = cbrt(r400384);
        double r400407 = r400406 * r400406;
        double r400408 = cbrt(r400391);
        double r400409 = r400408 * r400408;
        double r400410 = r400407 / r400409;
        double r400411 = r400405 * r400410;
        double r400412 = r400406 / r400408;
        double r400413 = fma(r400411, r400412, r400385);
        double r400414 = r400401 ? r400404 : r400413;
        double r400415 = r400399 ? r400393 : r400414;
        double r400416 = r400395 ? r400397 : r400415;
        return r400416;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.0
Target	8.1
Herbie	6.9

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 28.3

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

   3. Taylor expanded around inf 42.2

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

   4. Simplified 26.9

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

   if -inf.0 < (- (+ x y) (/ (* (- z t) y) (- a t))) < -1.0656173268700135e-277

   1. Initial program 1.3

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

   if -1.0656173268700135e-277 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 0.0

   1. Initial program 60.3

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

   2. Simplified 60.2

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

   4. Applied fma-udef 60.2

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

   6. Applied div-inv 60.2

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

   7. Applied associate-*l* 60.6

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

   8. Simplified 60.7

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

   10. Applied add-cube-cbrt 60.4

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

   11. Applied add-cube-cbrt 60.5

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

   12. Applied times-frac 60.4

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

   13. Applied associate-*r* 60.1

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

   14. Taylor expanded around inf 18.4

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

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

   1. Initial program 12.2

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

   2. Simplified 6.9

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

   4. Applied fma-udef 6.9

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

   6. Applied div-inv 6.9

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

   7. Applied associate-*l* 7.3

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

   8. Simplified 7.3

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

   10. Applied add-cube-cbrt 7.5

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

   11. Applied add-cube-cbrt 7.5

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

   12. Applied times-frac 7.5

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

   13. Applied associate-*r* 6.1

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

   15. Applied fma-def 6.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -1.065617326870013528561467193287073133332 \cdot 10^{-277}:\\ \;\;\;\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 0.0:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - z\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))