Average Error: 6.1 → 0.4

Time: 6.6min

Precision: binary64

Cost: 1730

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5.514422444088807 \cdot 10^{+257}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.3819062189176262 \cdot 10^{+287}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \leq -5.514422444088807 \cdot 10^{+257}:\\
\;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.3819062189176262 \cdot 10^{+287}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* y (- z t)) -5.514422444088807e+257)
   (+ x (* (- z t) (/ y a)))
   (if (<= (* y (- z t)) 1.3819062189176262e+287)
     (+ x (/ (* y (- z t)) a))
     (+ x (* y (/ (- z t) a))))))

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y * (z - t)) <= -5.514422444088807e+257) {
		tmp = x + ((z - t) * (y / a));
	} else if ((y * (z - t)) <= 1.3819062189176262e+287) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y * ((z - t) / a));
	}
	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	6.1
Target	0.7
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Alternatives

Alternative 1
Error	1.0
Cost	72000

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

Alternative 2
Error	1.0
Cost	39616

\[x + \frac{\frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}{z - t}}\]

Alternative 3
Error	1.0
Cost	39616

\[x + \frac{1}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{z - t}{\frac{\sqrt[3]{a}}{\sqrt[3]{y}}}\]

Alternative 4
Error	0.4
Cost	1416

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5.514422444088807 \cdot 10^{+257} \lor \neg \left(y \cdot \left(z - t\right) \leq 4.0307655758707046 \cdot 10^{+227}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Alternative 5
Error	2.3
Cost	576

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

Alternative 6
Error	2.4
Cost	576

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

Alternative 7
Error	8.8
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5564204691164272 \cdot 10^{+17} \lor \neg \left(t \leq 3.3737087326612843 \cdot 10^{-41}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 8
Error	8.8
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5564204691164272 \cdot 10^{+17} \lor \neg \left(t \leq 3.3845146637980385 \cdot 10^{-39}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array}\]

Alternative 9
Error	8.9
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4912506841300403 \cdot 10^{+17} \lor \neg \left(t \leq 1.5265949137561266 \cdot 10^{-41}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array}\]

Alternative 10
Error	9.7
Cost	776

\[\begin{array}{l} \mathbf{if}\;t \leq -0.002345680633920969 \lor \neg \left(t \leq 1.7644500327651093 \cdot 10^{-41}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array}\]

Alternative 11
Error	19.1
Cost	1739

\[\begin{array}{l} \mathbf{if}\;t \leq -9.769650233455533 \cdot 10^{+89}:\\ \;\;\;\;-\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq -2.3927873258594486 \cdot 10^{+45}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq -9.875110794919842 \cdot 10^{+29}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;t \leq 4.869973898575796 \cdot 10^{+100} \lor \neg \left(t \leq 3.881785513702398 \cdot 10^{+151}\right):\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t}{\frac{a}{y}}\\ \end{array}\]

Alternative 12
Error	28.0
Cost	1283

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0619953600607391 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.28337174311298 \cdot 10^{-269}:\\ \;\;\;\;-\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 6.930706779240972 \cdot 10^{-62}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 13
Error	28.3
Cost	1604

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2661162990789266 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.7335213432510176 \cdot 10^{-227}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.0403172204709538 \cdot 10^{-303}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;x \leq 1.7737819446138153 \cdot 10^{-61}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 14
Error	28.3
Cost	962

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7157302820236517 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.874203343316973 \cdot 10^{-62}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 15
Error	27.9
Cost	962

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7157302820236517 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7246350369454143 \cdot 10^{-61}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 16
Error	27.9
Cost	962

\[\begin{array}{l} \mathbf{if}\;x \leq -7.378909651655628 \cdot 10^{-76}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.110103578179451 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Alternative 17
Error	30.7
Cost	64

\[x\]

Derivation

Split input into 3 regimes
if (*.f64 y (-.f64 z t)) < -5.51442244408880728e257
1. Initial program 41.9
  \[x + \frac{y \cdot \left(z - t\right)}{a}\]
2. Using strategy rm
3. Applied add-cube-cbrt_binary64_1238942.2
  \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]
4. Applied associate-/r*_binary64_1229842.2
  \[\leadsto x + \color{blue}{\frac{\frac{y \cdot \left(z - t\right)}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}}\]
5. Simplified11.2
  \[\leadsto x + \frac{\color{blue}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}{\sqrt[3]{a}}\]
6. Using strategy rm
7. Applied clear-num_binary64_1235311.2
  \[\leadsto x + \color{blue}{\frac{1}{\frac{\sqrt[3]{a}}{\frac{y}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(z - t\right)}}}\]
8. Simplified0.3
  \[\leadsto x + \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}}\]
9. Using strategy rm
10. Applied div-inv_binary64_123510.3
  \[\leadsto x + \frac{1}{\color{blue}{\frac{a}{y} \cdot \frac{1}{z - t}}}\]
11. Applied add-sqr-sqrt_binary64_123760.3
  \[\leadsto x + \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{a}{y} \cdot \frac{1}{z - t}}\]
12. Applied times-frac_binary64_123600.3
  \[\leadsto x + \color{blue}{\frac{\sqrt{1}}{\frac{a}{y}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}}\]
13. Simplified0.2
  \[\leadsto x + \color{blue}{\frac{y}{a}} \cdot \frac{\sqrt{1}}{\frac{1}{z - t}}\]
14. Simplified0.2
  \[\leadsto x + \frac{y}{a} \cdot \color{blue}{\left(z - t\right)}\]
15. Simplified0.2
  \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(z - t\right)}\]
if -5.51442244408880728e257 < (*.f64 y (-.f64 z t)) < 1.3819062189176262e287
1. Initial program 0.4
  \[x + \frac{y \cdot \left(z - t\right)}{a}\]
2. Using strategy rm
3. Applied *-commutative_binary64_122850.4
  \[\leadsto x + \frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\]
4. Simplified0.4
  \[\leadsto \color{blue}{x + \frac{\left(z - t\right) \cdot y}{a}}\]
if 1.3819062189176262e287 < (*.f64 y (-.f64 z t))
1. Initial program 52.4
  \[x + \frac{y \cdot \left(z - t\right)}{a}\]
2. Using strategy rm
3. Applied *-un-lft-identity_binary64_1235452.4
  \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{1 \cdot a}}\]
4. Applied times-frac_binary64_123600.2
  \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - t}{a}}\]
5. Simplified0.2
  \[\leadsto x + \color{blue}{y} \cdot \frac{z - t}{a}\]
6. Simplified0.2
  \[\leadsto \color{blue}{x + y \cdot \frac{z - t}{a}}\]
Recombined 3 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5.514422444088807 \cdot 10^{+257}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.3819062189176262 \cdot 10^{+287}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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

Error

Try it out

Target

Alternatives

Error

Time

Derivation

`if (*.f64 y (-.f64 z t)) < -5.51442244408880728e257`

`if -5.51442244408880728e257 < (*.f64 y (-.f64 z t)) < 1.3819062189176262e287`

`if 1.3819062189176262e287 < (*.f64 y (-.f64 z t))`

Reproduce