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

Average Error: 10.7 → 1.4

Time: 8.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.46061279201978127 \cdot 10^{27}:\\ \;\;\;\;x + \left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}} \cdot \left(\sqrt[3]{t} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{{\left(\sqrt[3]{a - z}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t}} \cdot \left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;t \le 3.4738956651351627 \cdot 10^{75}:\\ \;\;\;\;x + \left(t \cdot \left(y - z\right)\right) \cdot \frac{1}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\sqrt[3]{a - z}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((t <= -3.460612792019781e+27)) {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) (y - z)) / ((double) pow(((double) cbrt(((double) (a - z)))), 2.0)))) * ((double) (((double) cbrt(t)) * ((double) (((double) cbrt(t)) / ((double) cbrt(((double) pow(((double) cbrt(((double) (a - z)))), 2.0)))))))))) * ((double) (((double) (((double) cbrt(((double) cbrt(t)))) * ((double) (((double) cbrt(((double) cbrt(t)))) * ((double) cbrt(((double) cbrt(t)))))))) / ((double) cbrt(((double) cbrt(((double) (a - z))))))))))));
	} else {
		double VAR_1;
		if ((t <= 3.4738956651351627e+75)) {
			VAR_1 = ((double) (x + ((double) (((double) (t * ((double) (y - z)))) * ((double) (1.0 / ((double) (a - z))))))));
		} else {
			VAR_1 = ((double) (x + ((double) (((double) (((double) (y - z)) / ((double) (((double) cbrt(((double) (a - z)))) * ((double) cbrt(((double) (a - z)))))))) * ((double) (t / ((double) cbrt(((double) (a - z))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.7
Target	0.6
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -3.46061279201978127e27

   1. Initial program 24.6

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

   2. Simplified 3.2

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

   4. Applied add-cube-cbrt 3.9

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

   5. Applied *-un-lft-identity 3.9

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

   6. Applied times-frac 3.9

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

   7. Applied associate-*r* 2.0

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

   8. Simplified 2.0

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t}{\sqrt[3]{a - z}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 2.0

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

   11. Applied cbrt-prod 2.1

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

   12. Applied add-cube-cbrt 2.2

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

   13. Applied times-frac 2.2

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

   14. Applied associate-*r* 1.3

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

   15. Simplified 1.3

       \[\leadsto x + \color{blue}{\left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{{\left(\sqrt[3]{a - z}\right)}^{2}}} \cdot \sqrt[3]{t}\right)\right)} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{a - z}}}\]
   16. Using strategy rm

   17. Applied add-cube-cbrt 1.4

       \[\leadsto x + \left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{{\left(\sqrt[3]{a - z}\right)}^{2}}} \cdot \sqrt[3]{t}\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right) \cdot \sqrt[3]{\sqrt[3]{t}}}}{\sqrt[3]{\sqrt[3]{a - z}}}\]

   if -3.46061279201978127e27 < t < 3.4738956651351627e75

   1. Initial program 1.1

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

   2. Simplified 3.0

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

   4. Applied div-inv 3.0

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

   5. Applied associate-*r* 1.2

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

   if 3.4738956651351627e75 < t

   1. Initial program 29.1

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

   2. Simplified 3.3

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

   4. Applied add-cube-cbrt 4.1

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

   5. Applied *-un-lft-identity 4.1

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

   6. Applied times-frac 4.1

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

   7. Applied associate-*r* 2.0

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

   8. Simplified 2.0

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.46061279201978127 \cdot 10^{27}:\\ \;\;\;\;x + \left(\frac{y - z}{{\left(\sqrt[3]{a - z}\right)}^{2}} \cdot \left(\sqrt[3]{t} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{{\left(\sqrt[3]{a - z}\right)}^{2}}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{t}} \cdot \left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;t \le 3.4738956651351627 \cdot 10^{75}:\\ \;\;\;\;x + \left(t \cdot \left(y - z\right)\right) \cdot \frac{1}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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