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

Average Error: 10.8 → 1.7

Time: 13.8s

Precision: binary64

Cost: 1032

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.266354716065063 \cdot 10^{-227} \lor \neg \left(x \leq 9.434183570764376 \cdot 10^{-09}\right):\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x 1.266354716065063e-227) (not (<= x 9.434183570764376e-09)))
   (+ x (* (/ (- y z) (- a z)) t))
   (+ x (/ (- y z) (/ (- a z) t)))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= 1.266354716065063e-227) || !(x <= 9.434183570764376e-09)) {
		tmp = x + (((y - z) / (a - z)) * t);
	} else {
		tmp = x + ((y - z) / ((a - z) / t));
	}
	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	10.8
Target	0.6
Herbie	1.7

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

Alternatives

Alternative 1
Error	1.2
Cost	40128

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

Alternative 2
Error	0.5
Cost	1032

\[\begin{array}{l} \mathbf{if}\;t \leq -2.607906820722103 \cdot 10^{+20} \lor \neg \left(t \leq 4.5875175130816466 \cdot 10^{-82}\right):\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \end{array}\]

Alternative 3
Error	1.5
Cost	1032

\[\begin{array}{l} \mathbf{if}\;t \leq -9.810270338854375 \cdot 10^{-119} \lor \neg \left(t \leq 2.121227199050854 \cdot 10^{-277}\right):\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \end{array}\]

Alternative 4
Error	10.9
Cost	2181

\[\begin{array}{l} \mathbf{if}\;z \leq -2.269628994657719 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -1.8891242327213466 \cdot 10^{-49}:\\ \;\;\;\;x - z \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 2.9658245924063643 \cdot 10^{-104}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \mathbf{elif}\;z \leq 1.7171850962601488 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 2.7399348652501344 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array}\]

Alternative 5
Error	10.9
Cost	2181

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3744400148381077 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -2.993230862827396 \cdot 10^{-49}:\\ \;\;\;\;x - z \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 2.9658245924063643 \cdot 10^{-104}:\\ \;\;\;\;x + y \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 1.7171850962601488 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 1.6915671475104914 \cdot 10^{+52}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array}\]

Alternative 6
Error	10.6
Cost	1539

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5916010666957924 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq -3.0829907836386377 \cdot 10^{-50}:\\ \;\;\;\;x - z \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 4.005831574744566 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array}\]

Alternative 7
Error	12.5
Cost	1860

\[\begin{array}{l} \mathbf{if}\;a \leq -1.7810524236303283 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq -220.647905604936:\\ \;\;\;\;x - z \cdot \frac{t}{a - z}\\ \mathbf{elif}\;a \leq -3.196925398238006 \cdot 10^{-93}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 3.0339013843065053 \cdot 10^{-09}:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \end{array}\]

Alternative 8
Error	12.8
Cost	1860

\[\begin{array}{l} \mathbf{if}\;a \leq -5.665548523982956 \cdot 10^{+45}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq -144.80765736241221:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{elif}\;a \leq -3.196925398238006 \cdot 10^{-93}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 1.8481797992319505 \cdot 10^{-09}:\\ \;\;\;\;x - \frac{y - z}{\frac{z}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \end{array}\]

Alternative 9
Error	15.0
Cost	1860

\[\begin{array}{l} \mathbf{if}\;a \leq -7.205370870232716 \cdot 10^{+45}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \mathbf{elif}\;a \leq -226979239.8244441:\\ \;\;\;\;x + t\\ \mathbf{elif}\;a \leq -4.219313419649227 \cdot 10^{-94}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 4.666143669536909 \cdot 10^{+34}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a}{t}}\\ \end{array}\]

Alternative 10
Error	14.2
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -2.539804671575186 \cdot 10^{-41} \lor \neg \left(z \leq 3.8392687687523886 \cdot 10^{+50}\right):\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{t}}\\ \end{array}\]

Alternative 11
Error	19.0
Cost	1041

\[\begin{array}{l} \mathbf{if}\;a \leq -4.623624693564985 \cdot 10^{+150} \lor \neg \left(a \leq -7928054331.279097 \lor \neg \left(a \leq -4.640645079994026 \cdot 10^{-97}\right) \land a \leq 4.082774562727311 \cdot 10^{+33}\right):\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array}\]

Alternative 12
Error	22.9
Cost	908

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1708911965137315 \cdot 10^{+128} \lor \neg \left(y \leq -4.471136920446894 \cdot 10^{+69}\right) \land y \leq 7.609833274098096 \cdot 10^{+170}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a - z}\\ \end{array}\]

Alternative 13
Error	20.9
Cost	513

\[\begin{array}{l} \mathbf{if}\;a \leq -8.231696246658259 \cdot 10^{+140}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array}\]

Alternative 14
Error	28.7
Cost	64

\[x\]

Alternative 15
Error	61.9
Cost	64

\[-1\]

Alternative 16
Error	61.8
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if x < 1.2663547160650629e-227 or 9.4341835707643759e-9 < x

   1. Initial program 10.8

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

   3. Applied associate-/l*_binary64_17073 2.9

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

   5. Applied associate-/r/_binary64_17074 1.2

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

   6. Simplified 1.2

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

   if 1.2663547160650629e-227 < x < 9.4341835707643759e-9

   1. Initial program 10.5

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

   3. Applied associate-/l*_binary64_17073 3.8

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

   4. Simplified 3.8

      \[\leadsto \color{blue}{x + \frac{y - z}{\frac{a - z}{t}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.266354716065063 \cdot 10^{-227} \lor \neg \left(x \leq 9.434183570764376 \cdot 10^{-09}\right):\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))