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

Average Error: 16.0 → 8.3

Time: 23.4s

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 r27279051 = x;
        double r27279052 = y;
        double r27279053 = r27279051 + r27279052;
        double r27279054 = z;
        double r27279055 = t;
        double r27279056 = r27279054 - r27279055;
        double r27279057 = r27279056 * r27279052;
        double r27279058 = a;
        double r27279059 = r27279058 - r27279055;
        double r27279060 = r27279057 / r27279059;
        double r27279061 = r27279053 - r27279060;
        return r27279061;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r27279062 = a;
        double r27279063 = -6.796987859803041e-120;
        bool r27279064 = r27279062 <= r27279063;
        double r27279065 = x;
        double r27279066 = y;
        double r27279067 = r27279065 + r27279066;
        double r27279068 = cbrt(r27279066);
        double r27279069 = t;
        double r27279070 = r27279062 - r27279069;
        double r27279071 = cbrt(r27279070);
        double r27279072 = r27279068 / r27279071;
        double r27279073 = z;
        double r27279074 = r27279069 - r27279073;
        double r27279075 = r27279071 * r27279071;
        double r27279076 = r27279074 / r27279075;
        double r27279077 = r27279068 * r27279068;
        double r27279078 = r27279076 * r27279077;
        double r27279079 = r27279072 * r27279078;
        double r27279080 = r27279067 + r27279079;
        double r27279081 = 2.976075834772394e-120;
        bool r27279082 = r27279062 <= r27279081;
        double r27279083 = r27279073 / r27279069;
        double r27279084 = fma(r27279083, r27279066, r27279065);
        double r27279085 = r27279066 / r27279071;
        double r27279086 = fma(r27279076, r27279085, r27279067);
        double r27279087 = r27279082 ? r27279084 : r27279086;
        double r27279088 = r27279064 ? r27279080 : r27279087;
        return r27279088;
}

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))))