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

Average Error: 16.0 → 8.3

Time: 25.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.796987859803041 \cdot 10^{-120}:\\ \;\;\;\;\left(x + y\right) + \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\\ \mathbf{elif}\;a \le 2.976075834772394 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{y}{\sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r15579401 = x;
        double r15579402 = y;
        double r15579403 = r15579401 + r15579402;
        double r15579404 = z;
        double r15579405 = t;
        double r15579406 = r15579404 - r15579405;
        double r15579407 = r15579406 * r15579402;
        double r15579408 = a;
        double r15579409 = r15579408 - r15579405;
        double r15579410 = r15579407 / r15579409;
        double r15579411 = r15579403 - r15579410;
        return r15579411;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r15579412 = a;
        double r15579413 = -6.796987859803041e-120;
        bool r15579414 = r15579412 <= r15579413;
        double r15579415 = x;
        double r15579416 = y;
        double r15579417 = r15579415 + r15579416;
        double r15579418 = cbrt(r15579416);
        double r15579419 = t;
        double r15579420 = r15579412 - r15579419;
        double r15579421 = cbrt(r15579420);
        double r15579422 = r15579418 / r15579421;
        double r15579423 = z;
        double r15579424 = r15579419 - r15579423;
        double r15579425 = r15579421 * r15579421;
        double r15579426 = r15579424 / r15579425;
        double r15579427 = r15579418 * r15579418;
        double r15579428 = r15579426 * r15579427;
        double r15579429 = r15579422 * r15579428;
        double r15579430 = r15579417 + r15579429;
        double r15579431 = 2.976075834772394e-120;
        bool r15579432 = r15579412 <= r15579431;
        double r15579433 = r15579423 / r15579419;
        double r15579434 = fma(r15579433, r15579416, r15579415);
        double r15579435 = r15579416 / r15579421;
        double r15579436 = fma(r15579426, r15579435, r15579417);
        double r15579437 = r15579432 ? r15579434 : r15579436;
        double r15579438 = r15579414 ? r15579430 : r15579437;
        return r15579438;
}

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	8.3

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -6.796987859803041e-120

   1. Initial program 14.2

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

   2. Simplified 8.2

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

   4. Applied fma-udef 8.2

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

   6. Applied add-cube-cbrt 8.4

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

   7. Applied *-un-lft-identity 8.4

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

   8. Applied times-frac 8.4

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

   9. Applied associate-*r* 7.8

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

   10. Simplified 7.8

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

   12. Applied *-un-lft-identity 7.8

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

   13. Applied cbrt-prod 7.8

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

   14. Applied add-cube-cbrt 7.8

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

   15. Applied times-frac 7.8

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

   16. Applied associate-*r* 7.5

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

   17. Simplified 7.5

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

   if -6.796987859803041e-120 < a < 2.976075834772394e-120

   1. Initial program 19.8

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

   2. Simplified 19.6

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

   3. Taylor expanded around inf 10.3

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

   4. Simplified 9.1

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

   if 2.976075834772394e-120 < a

   1. Initial program 14.7

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

   2. Simplified 8.9

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

   4. Applied fma-udef 9.0

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied *-un-lft-identity 9.1

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

   8. Applied times-frac 9.1

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

   9. Applied associate-*r* 8.5

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

   10. Simplified 8.5

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

   12. Applied fma-def 8.5

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.796987859803041 \cdot 10^{-120}:\\ \;\;\;\;\left(x + y\right) + \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \left(\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\\ \mathbf{elif}\;a \le 2.976075834772394 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{y}{\sqrt[3]{a - t}}, x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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 (- 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))))