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

Average Error: 16.1 → 9.9

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -6.671541700787339 \cdot 10^{+222}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -2.5969526405097103 \cdot 10^{+82}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;t \leq -6.436488181590957 \cdot 10^{+42} \lor \neg \left(t \leq 6.426053483705132 \cdot 10^{+76}\right):\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \left(\frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{\sqrt[3]{z - t}}{{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -6.671541700787339e+222)
   (+ x (/ (* z y) t))
   (if (<= t -2.5969526405097103e+82)
     (- (+ x y) (/ (- z t) (/ (- a t) y)))
     (if (or (<= t -6.436488181590957e+42) (not (<= t 6.426053483705132e+76)))
       (+ x (/ (* z y) t))
       (-
        (+ x y)
        (*
         (/
          (* (cbrt (- z t)) (cbrt (- z t)))
          (* (cbrt (cbrt (cbrt (- a t)))) (cbrt (cbrt (cbrt (- a t))))))
         (*
          (/ y (cbrt (cbrt (cbrt (- a t)))))
          (/
           (cbrt (- z t))
           (pow (cbrt (* (cbrt (- a t)) (cbrt (- a t)))) 4.0)))))))))

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (x + y)) - (((double) (((double) (z - t)) * y)) / ((double) (a - t)))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6.671541700787339e+222)) {
		tmp = ((double) (x + (((double) (z * y)) / t)));
	} else {
		double tmp_1;
		if ((t <= -2.5969526405097103e+82)) {
			tmp_1 = ((double) (((double) (x + y)) - (((double) (z - t)) / (((double) (a - t)) / y))));
		} else {
			double tmp_2;
			if (((t <= -6.436488181590957e+42) || !(t <= 6.426053483705132e+76))) {
				tmp_2 = ((double) (x + (((double) (z * y)) / t)));
			} else {
				tmp_2 = ((double) (((double) (x + y)) - ((double) ((((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))) / ((double) (((double) cbrt(((double) cbrt(((double) cbrt(((double) (a - t)))))))) * ((double) cbrt(((double) cbrt(((double) cbrt(((double) (a - t))))))))))) * ((double) ((y / ((double) cbrt(((double) cbrt(((double) cbrt(((double) (a - t))))))))) * (((double) cbrt(((double) (z - t)))) / ((double) pow(((double) cbrt(((double) (((double) cbrt(((double) (a - t)))) * ((double) cbrt(((double) (a - t)))))))), 4.0)))))))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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	9.9

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

Derivation

1. Split input into 3 regimes

2. if t < -6.6715417007873388e222 or -2.5969526405097103e82 < t < -6.43648818159095696e42 or 6.42605348370513173e76 < t

   1. Initial program 29.8

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

   2. Taylor expanded around inf 17.1

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

   3. Simplified 17.1

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

   if -6.6715417007873388e222 < t < -2.5969526405097103e82

   1. Initial program 24.7

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

   3. Applied associate-/l*_binary64 16.2

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

   if -6.43648818159095696e42 < t < 6.42605348370513173e76

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt_binary64 7.2

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

   4. Applied times-frac_binary64 5.1

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

   6. Applied add-cube-cbrt_binary64 5.1

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

   7. Applied cbrt-prod_binary64 5.2

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

   8. Applied *-un-lft-identity_binary64 5.2

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

   9. Applied times-frac_binary64 5.2

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

   10. Applied associate-*r*_binary64 5.4

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

   11. Simplified 5.4

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

   13. Applied add-cube-cbrt_binary64 5.4

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

   14. Applied *-un-lft-identity_binary64 5.4

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

   15. Applied times-frac_binary64 5.5

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

   16. Applied associate-*r*_binary64 5.5

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

   17. Simplified 5.6

       \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\]
   18. Using strategy rm

   19. Applied add-cube-cbrt_binary64 5.6

       \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\]

   20. Applied times-frac_binary64 5.6

       \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{\sqrt[3]{z - t}}{{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}}\right)} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\]

   21. Applied associate-*l*_binary64 4.7

       \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \left(\frac{\sqrt[3]{z - t}}{{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}}\right)}\]

   22. Simplified 4.7

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.671541700787339 \cdot 10^{+222}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -2.5969526405097103 \cdot 10^{+82}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \mathbf{elif}\;t \leq -6.436488181590957 \cdot 10^{+42} \lor \neg \left(t \leq 6.426053483705132 \cdot 10^{+76}\right):\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \left(\frac{y}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - t}}}} \cdot \frac{\sqrt[3]{z - t}}{{\left(\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right)}^{4}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))