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

Average Error: 10.2 → 0.4

Time: 12.8s

Precision: binary64

Cost: 2370

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;x + y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 2.482279323724117 \cdot 10^{+282}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (* y (- z t)) (- z a)) (- INFINITY))
   (+ x (* y (* (- z t) (/ 1.0 (- z a)))))
   (if (<= (/ (* y (- z t)) (- z a)) 2.482279323724117e+282)
     (+ (/ (* y (- z t)) (- z a)) x)
     (+ x (* (- z t) (/ y (- z a)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y * (z - t)) / (z - a)) <= -((double) INFINITY)) {
		tmp = x + (y * ((z - t) * (1.0 / (z - a))));
	} else if (((y * (z - t)) / (z - a)) <= 2.482279323724117e+282) {
		tmp = ((y * (z - t)) / (z - a)) + x;
	} else {
		tmp = x + ((z - t) * (y / (z - a)));
	}
	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	10.2
Target	1.3
Herbie	0.4

\[x + \frac{y}{\frac{z - a}{z - t}}\]

Alternatives

Alternative 1
Error	0.8
Cost	59584

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

Alternative 2
Error	1.1
Cost	39872

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

Alternative 3
Error	0.4
Cost	2370

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 2.482279323724117 \cdot 10^{+282}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array}\]

Alternative 4
Error	0.2
Cost	2370

\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 3.12498673656695 \cdot 10^{+307}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

Alternative 5
Error	1.8
Cost	1032

\[\begin{array}{l} \mathbf{if}\;y \leq -4.591716408584272 \cdot 10^{-64} \lor \neg \left(y \leq 1.1020012389785992 \cdot 10^{-221}\right):\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot t}{z - a}\\ \end{array}\]

Alternative 6
Error	7.8
Cost	1218

\[\begin{array}{l} \mathbf{if}\;t \leq -1.367391352156437 \cdot 10^{+55}:\\ \;\;\;\;x - y \cdot \frac{t}{z - a}\\ \mathbf{elif}\;t \leq 2316768892041.1904:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array}\]

Alternative 7
Error	8.2
Cost	904

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1208769394479888 \cdot 10^{+56} \lor \neg \left(t \leq 70161090145.47116\right):\\ \;\;\;\;x - y \cdot \frac{t}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1 - \frac{a}{z}}\\ \end{array}\]

Alternative 8
Error	9.1
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -9.097515045328543 \cdot 10^{+103} \lor \neg \left(z \leq 5.002842356276485 \cdot 10^{+61}\right):\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{t}{z - a}\\ \end{array}\]

Alternative 9
Error	14.4
Cost	1860

\[\begin{array}{l} \mathbf{if}\;a \leq -1.9727090101242836 \cdot 10^{-35}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;a \leq 4.2591287773618582 \cdot 10^{-106}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;a \leq 3.3569483171766777 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 7.369457564098547 \cdot 10^{+139}:\\ \;\;\;\;x + y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array}\]

Alternative 10
Error	17.0
Cost	1041

\[\begin{array}{l} \mathbf{if}\;a \leq -4.0271371946744994 \cdot 10^{-69} \lor \neg \left(a \leq 1.7241342036448077 \cdot 10^{+18} \lor \neg \left(a \leq 2.313510324046897 \cdot 10^{+74}\right) \land a \leq 5.633727489378703 \cdot 10^{+184}\right):\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array}\]

Alternative 11
Error	19.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;z \leq -8.635899568710024 \cdot 10^{-72} \lor \neg \left(z \leq 2.933537414866347 \cdot 10^{-113}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 12
Error	29.1
Cost	64

\[x\]

Alternative 13
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/l*_binary64_16732 0.1

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

   5. Applied div-inv_binary64_16784 0.1

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

   6. Simplified 0.2

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

   7. Simplified 0.2

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 2.482279323724117e282

   1. Initial program 0.2

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

   2. Simplified 0.2

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

   if 2.482279323724117e282 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 58.9

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

   3. Applied associate-/l*_binary64_16732 0.8

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

   5. Applied associate-/r/_binary64_16733 1.9

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

   6. Simplified 1.9

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;x + y \cdot \left(\left(z - t\right) \cdot \frac{1}{z - a}\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 2.482279323724117 \cdot 10^{+282}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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