Average Error: 1.5 → 0.6

Time: 12.2s

Precision: binary64

Cost: 1474

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.960308320085401 \cdot 10^{-20}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 1.672021968335163 \cdot 10^{+43}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

x + y \cdot \frac{z - t}{z - a}

↓

\begin{array}{l}
\mathbf{if}\;y \leq -5.960308320085401 \cdot 10^{-20}:\\
\;\;\;\;x + y \cdot \frac{z - t}{z - a}\\

\mathbf{elif}\;y \leq 1.672021968335163 \cdot 10^{+43}:\\
\;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{z - a}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\

\end{array}

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5.960308320085401e-20)
   (+ x (* y (/ (- z t) (- z a))))
   (if (<= y 1.672021968335163e+43)
     (+ x (* (* y (- z t)) (/ 1.0 (- z a))))
     (+ x (/ y (/ (- z a) (- z t)))))))

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 <= -5.960308320085401e-20) {
		tmp = x + (y * ((z - t) / (z - a)));
	} else if (y <= 1.672021968335163e+43) {
		tmp = x + ((y * (z - t)) * (1.0 / (z - a)));
	} else {
		tmp = x + (y / ((z - a) / (z - t)));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	1.5
Target	1.4
Herbie	0.6

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

Alternatives

Alternative 1
Error	0.6
Cost	40128

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

Alternative 2
Error	0.6
Cost	1346

\[\begin{array}{l} \mathbf{if}\;y \leq -2.45465729562505 \cdot 10^{-60}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 1.672021968335163 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

Alternative 3
Error	0.9
Cost	1409

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -8.356417927164571 \cdot 10^{+235}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array}\]

Alternative 4
Error	0.9
Cost	1409

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1.1791595430829916 \cdot 10^{+168}:\\ \;\;\;\;x - \frac{t}{\frac{z - a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

Alternative 5
Error	2.2
Cost	2691

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1.0620475210778985 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{t}{\frac{z - a}{y}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1.1106907187803091 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array}\]

Alternative 6
Error	2.4
Cost	2691

\[\begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -1.0620475210778985 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{t}{\frac{z - a}{y}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 7.578109156794213 \cdot 10^{-11}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 1:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array}\]

Alternative 7
Error	12.4
Cost	1169

\[\begin{array}{l} \mathbf{if}\;z \leq -2.184583787047839 \cdot 10^{+98} \lor \neg \left(z \leq -7.1804636252200155 \cdot 10^{-12} \lor \neg \left(z \leq -8.142568326833419 \cdot 10^{-169}\right) \land z \leq 2.3126298001576838 \cdot 10^{+55}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{z - a}\\ \end{array}\]

Alternative 8
Error	14.8
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -6.598857573502581 \cdot 10^{-169} \lor \neg \left(z \leq 2.930854184528944 \cdot 10^{+52}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Alternative 9
Error	16.0
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -8.142568326833419 \cdot 10^{-169} \lor \neg \left(z \leq 1.7508724934294406 \cdot 10^{-91}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array}\]

Alternative 10
Error	15.5
Cost	776

\[\begin{array}{l} \mathbf{if}\;z \leq -8.142568326833419 \cdot 10^{-169} \lor \neg \left(z \leq 2.4940199285885987 \cdot 10^{+51}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array}\]

Alternative 11
Error	21.9
Cost	192

\[y + x\]

Alternative 12
Error	28.6
Cost	64

\[x\]

Alternative 13
Error	61.8
Cost	64

\[1\]

Derivation

Split input into 3 regimes
if y < -5.9603083200854013e-20
1. Initial program 0.6
  \[x + y \cdot \frac{z - t}{z - a}\]
2. Simplified0.6
  \[\leadsto \color{blue}{x + y \cdot \frac{z - t}{z - a}}\]
if -5.9603083200854013e-20 < y < 1.6720219683351629e43
1. Initial program 2.2
  \[x + y \cdot \frac{z - t}{z - a}\]
2. Using strategy rm
3. Applied div-inv_binary64_113282.2
  \[\leadsto x + y \cdot \color{blue}{\left(\left(z - t\right) \cdot \frac{1}{z - a}\right)}\]
4. Applied associate-*r*_binary64_112710.4
  \[\leadsto x + \color{blue}{\left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{z - a}}\]
5. Simplified0.4
  \[\leadsto \color{blue}{x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{z - a}}\]
if 1.6720219683351629e43 < y
1. Initial program 0.8
  \[x + y \cdot \frac{z - t}{z - a}\]
2. Using strategy rm
3. Applied clear-num_binary64_113301.0
  \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{z - a}{z - t}}}\]
4. Using strategy rm
5. Applied un-div-inv_binary64_113290.9
  \[\leadsto x + \color{blue}{\frac{y}{\frac{z - a}{z - t}}}\]
6. Simplified0.9
  \[\leadsto \color{blue}{x + \frac{y}{\frac{z - a}{z - t}}}\]
Recombined 3 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.960308320085401 \cdot 10^{-20}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 1.672021968335163 \cdot 10^{+43}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{z - a}{z - t}}\\ \end{array}\]

Reproduce

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

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

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

Error

Try it out

Target

Alternatives

Error

Derivation

`if y < -5.9603083200854013e-20`

`if -5.9603083200854013e-20 < y < 1.6720219683351629e43`

`if 1.6720219683351629e43 < y`

Reproduce