Average Error: 1.5 → 0.2

Time: 8.7s

Precision: binary64

Cost: 9026

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8755087116004516 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.7579917362672096 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\\ \end{array}\]

\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|

↓

\begin{array}{l}
\mathbf{if}\;x \leq -2.8755087116004516 \cdot 10^{-15}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \leq 4.7579917362672096 \cdot 10^{+24}:\\
\;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\\

\end{array}

(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8755087116004516e-15)
   (fabs (- (/ (+ x 4.0) y) (* x (/ z y))))
   (if (<= x 4.7579917362672096e+24)
     (fabs
      (/
       (* (- (/ x y) (/ 4.0 y)) (- (* y (+ (/ x y) (/ 4.0 y))) (* x z)))
       (* y (- (/ x y) (/ 4.0 y)))))
     (fabs (- (+ (/ x y) (/ 4.0 y)) (* z (/ x y)))))))

double code(double x, double y, double z) {
	return fabs(((x + 4.0) / y) - ((x / y) * z));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8755087116004516e-15) {
		tmp = fabs(((x + 4.0) / y) - (x * (z / y)));
	} else if (x <= 4.7579917362672096e+24) {
		tmp = fabs((((x / y) - (4.0 / y)) * ((y * ((x / y) + (4.0 / y))) - (x * z))) / (y * ((x / y) - (4.0 / y))));
	} else {
		tmp = fabs(((x / y) + (4.0 / y)) - (z * (x / y)));
	}
	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

Alternatives

Alternative 1
Error	0.7
Cost	46016

\[\left|\frac{x + 4}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right|\]

Alternative 2
Error	32.5
Cost	39488

\[\left|\frac{x + 4}{y} - \frac{\sqrt{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{\sqrt{x}}{\sqrt[3]{y}}\right)\right|\]

Alternative 3
Error	32.1
Cost	39488

\[\left|\frac{x + 4}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt{y}} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt{y}}\right)\right|\]

Alternative 4
Error	2.5
Cost	27072

\[\left|\sqrt[3]{\frac{x + 4}{y}} \cdot \left(\sqrt[3]{\frac{x + 4}{y}} \cdot \sqrt[3]{\frac{x + 4}{y}}\right) - z \cdot \frac{x}{y}\right|\]

Alternative 5
Error	1.8
Cost	27072

\[\left|\frac{x + 4}{y} - \sqrt[3]{z \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{z \cdot \frac{x}{y}} \cdot \sqrt[3]{z \cdot \frac{x}{y}}\right)\right|\]

Alternative 6
Error	2.5
Cost	26816

\[\left|\left(\sqrt[3]{x + 4} \cdot \sqrt[3]{x + 4}\right) \cdot \frac{\sqrt[3]{x + 4}}{y} - z \cdot \frac{x}{y}\right|\]

Alternative 7
Error	1.8
Cost	26816

\[\left|\frac{x + 4}{y} - \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \left(z \cdot \sqrt[3]{\frac{x}{y}}\right)\right|\]

Alternative 8
Error	1.8
Cost	26560

\[\left|\frac{x + 4}{y} - \sqrt[3]{z} \cdot \left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right|\]

Alternative 9
Error	2.7
Cost	26560

\[\left|\frac{x + 4}{y} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(z \cdot \frac{\sqrt[3]{x}}{y}\right)\right|\]

Alternative 10
Error	13.8
Cost	20160

\[\left|\sqrt{x + 4} \cdot \frac{\sqrt{x + 4}}{y} - z \cdot \frac{x}{y}\right|\]

Alternative 11
Error	29.1
Cost	20160

\[\left|\frac{x + 4}{y} - \sqrt{\frac{x}{y}} \cdot \left(z \cdot \sqrt{\frac{x}{y}}\right)\right|\]

Alternative 12
Error	33.4
Cost	20032

\[\left|\frac{x + 4}{y} - \sqrt{x} \cdot \left(z \cdot \frac{\sqrt{x}}{y}\right)\right|\]

Alternative 13
Error	16.9
Cost	14080

\[\left|\frac{{x}^{3} + 64}{y \cdot \left(16 + x \cdot \left(x - 4\right)\right)} - z \cdot \frac{x}{y}\right|\]

Alternative 14
Error	10.2
Cost	8384

\[\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\]

Alternative 15
Error	18.4
Cost	8000

\[\left|\frac{\left(x + 4\right) \cdot \left(x + 4\right) - \left(x \cdot z\right) \cdot \left(x \cdot z\right)}{y \cdot \left(\left(x + 4\right) + x \cdot z\right)}\right|\]

Alternative 16
Error	12.7
Cost	7488

\[\left|\frac{x \cdot x - 16}{y \cdot \left(x - 4\right)} - z \cdot \frac{x}{y}\right|\]

Alternative 17
Error	1.5
Cost	7232

\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\]

Alternative 18
Error	1.6
Cost	7232

\[\left|\left(x + 4\right) \cdot \frac{1}{y} - z \cdot \frac{x}{y}\right|\]

Alternative 19
Error	1.6
Cost	7232

\[\left|\frac{1}{\frac{y}{x + 4}} - z \cdot \frac{x}{y}\right|\]

Alternative 20
Error	3.4
Cost	7104

\[\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\]

Alternative 21
Error	1.5
Cost	7104

\[\left|\frac{x + 4}{y} - z \cdot \frac{x}{y}\right|\]

Alternative 22
Error	3.4
Cost	6976

\[\left|\frac{x + \left(4 - x \cdot z\right)}{y}\right|\]

Alternative 23
Error	16.1
Cost	6976

\[\left|\frac{4}{y} - z \cdot \frac{x}{y}\right|\]

Alternative 24
Error	3.4
Cost	6976

\[\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\]

Alternative 25
Error	18.2
Cost	6848

\[\left|\frac{x}{y} + \frac{4}{y}\right|\]

Alternative 26
Error	31.0
Cost	6848

\[\left|\frac{x}{y} \cdot \left(1 - z\right)\right|\]

Alternative 27
Error	33.0
Cost	6848

\[\left|\frac{x - x \cdot z}{y}\right|\]

Alternative 28
Error	45.1
Cost	6784

\[\left|-z \cdot \frac{x}{y}\right|\]

Alternative 29
Error	47.1
Cost	6784

\[\left|\frac{-x \cdot z}{y}\right|\]

Alternative 30
Error	18.2
Cost	6720

\[\left|\frac{x + 4}{y}\right|\]

Alternative 31
Error	32.4
Cost	6592

\[\left|\frac{4}{y}\right|\]

Alternative 32
Error	60.5
Cost	64

\[1\]

Alternative 33
Error	61.9
Cost	64

\[0\]

Alternative 34
Error	63.0
Cost	64

\[-1\]

Derivation

Split input into 3 regimes
if x < -2.8755087116004516e-15
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Using strategy rm
3. Applied div-inv_binary64_4160.2
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]
4. Applied associate-*l*_binary64_3600.1
  \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]
5. Simplified0.1
  \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
6. Simplified0.1
  \[\leadsto \color{blue}{\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|}\]
if -2.8755087116004516e-15 < x < 4.75799173626720961e24
1. Initial program 2.4
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around 0 2.4
  \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right)} - \frac{x}{y} \cdot z\right|\]
3. Simplified2.4
  \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + \frac{4}{y}\right)} - \frac{x}{y} \cdot z\right|\]
4. Using strategy rm
5. Applied associate-*l/_binary64_3620.2
  \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - \color{blue}{\frac{x \cdot z}{y}}\right|\]
6. Applied flip-+_binary64_39325.7
  \[\leadsto \left|\color{blue}{\frac{\frac{x}{y} \cdot \frac{x}{y} - \frac{4}{y} \cdot \frac{4}{y}}{\frac{x}{y} - \frac{4}{y}}} - \frac{x \cdot z}{y}\right|\]
7. Applied frac-sub_binary64_42825.8
  \[\leadsto \left|\color{blue}{\frac{\left(\frac{x}{y} \cdot \frac{x}{y} - \frac{4}{y} \cdot \frac{4}{y}\right) \cdot y - \left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(x \cdot z\right)}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot y}}\right|\]
8. Simplified0.3
  \[\leadsto \left|\frac{\color{blue}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}}{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot y}\right|\]
9. Simplified0.3
  \[\leadsto \left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{\color{blue}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}}\right|\]
10. Simplified0.3
  \[\leadsto \color{blue}{\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|}\]
if 4.75799173626720961e24 < x
1. Initial program 0.1
  \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
2. Taylor expanded around 0 0.1
  \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right)} - \frac{x}{y} \cdot z\right|\]
3. Simplified0.1
  \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + \frac{4}{y}\right)} - \frac{x}{y} \cdot z\right|\]
4. Simplified0.1
  \[\leadsto \color{blue}{\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|}\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8755087116004516 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \leq 4.7579917362672096 \cdot 10^{+24}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(y \cdot \left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))

Error

Try it out

Alternatives

Error

Derivation

`if x < -2.8755087116004516e-15`

`if -2.8755087116004516e-15 < x < 4.75799173626720961e24`

`if 4.75799173626720961e24 < x`

Reproduce