Average Error: 0.1 → 0.1

Time: 8.8s

Precision: binary64

Cost: 7680

\[0 < m \land 0 < v \land v < 0.25\]

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]

↓

\[\frac{{m}^{3}}{v} + \left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right)\]

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)

↓

\frac{{m}^{3}}{v} + \left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right)

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

↓

(FPCore (m v)
 :precision binary64
 (+ (/ (pow m 3.0) v) (- (+ m (/ m v)) (+ 1.0 (* 2.0 (* m (/ m v)))))))

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

↓

double code(double m, double v) {
	return (pow(m, 3.0) / v) + ((m + (m / v)) - (1.0 + (2.0 * (m * (m / v)))));
}

Error

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

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	0.8
Cost	21824

\[\sqrt[3]{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \cdot \left(\sqrt[3]{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \cdot \sqrt[3]{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}\right)\]

Alternative 2
Error	0.9
Cost	20288

\[\left(1 - m\right) \cdot \left(\frac{m}{\sqrt[3]{v} \cdot \sqrt[3]{v}} \cdot \frac{1 - m}{\sqrt[3]{v}} - 1\right)\]

Alternative 3
Error	28.0
Cost	14528

\[\sqrt{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \cdot \sqrt{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}\]

Alternative 4
Error	9.6
Cost	14144

\[\left(1 - m\right) \cdot \left(\sqrt{\frac{m \cdot \left(1 - m\right)}{v}} \cdot \sqrt{\frac{m \cdot \left(1 - m\right)}{v}} - 1\right)\]

Alternative 5
Error	0.1
Cost	13888

\[\left(\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 + \sqrt{m}\right)\right) \cdot \left(1 - \sqrt{m}\right)\]

Alternative 6
Error	0.4
Cost	13760

\[\left(1 - m\right) \cdot \left(\frac{m}{\sqrt{v}} \cdot \frac{1 - m}{\sqrt{v}} - 1\right)\]

Alternative 7
Error	20.2
Cost	13696

\[\left(1 - m\right) \cdot \left(-1 + \sqrt[3]{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3}}\right)\]

Alternative 8
Error	0.1
Cost	7808

\[\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right) + \frac{1}{\frac{v}{{m}^{3}}}\]

Alternative 9
Error	3.5
Cost	7808

\[\frac{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - {m}^{3}\right)}{1 + \left(m + m \cdot m\right)}\]

Alternative 10
Error	28.3
Cost	7552

\[\left(m + \frac{m}{v}\right) + \left(\frac{{m}^{3}}{v} - 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\]

Alternative 11
Error	53.8
Cost	6656

\[\frac{{m}^{3}}{v}\]

Alternative 12
Error	13.6
Cost	2368

\[\frac{\left(1 - m \cdot m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v}\right)}{\left(m + 1\right) \cdot \left(1 + \frac{m \cdot \left(1 - m\right)}{v}\right)}\]

Alternative 13
Error	0.1
Cost	1728

\[\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right) + \frac{1}{\frac{1}{m \cdot m} \cdot \frac{v}{m}}\]

Alternative 14
Error	0.1
Cost	1600

\[\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right) + \frac{1}{\frac{\frac{v}{m \cdot m}}{m}}\]

Alternative 15
Error	0.1
Cost	1344

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + m \cdot \left(1 - \frac{m \cdot \left(1 - m\right)}{v}\right)\]

Alternative 16
Error	0.1
Cost	1216

\[\left(1 - m\right) \cdot \left(-1 + \frac{m \cdot \left(1 - m \cdot m\right)}{v \cdot \left(m + 1\right)}\right)\]

Alternative 17
Error	0.2
Cost	960

\[\left(1 - m\right) \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v} - 1\right)\]

Alternative 18
Error	0.2
Cost	960

\[\left(1 - m\right) \cdot \left(\frac{1}{\frac{v}{m \cdot \left(1 - m\right)}} - 1\right)\]

Alternative 19
Error	0.2
Cost	832

\[\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v} - 1\right)\]

Alternative 20
Error	0.1
Cost	832

\[\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)\]

Alternative 21
Error	0.1
Cost	832

\[\left(1 - m\right) \cdot \left(-1 + \frac{m}{\frac{v}{1 - m}}\right)\]

Alternative 22
Error	28.4
Cost	704

\[\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\]

Alternative 23
Error	29.1
Cost	704

\[\left(1 - m\right) \cdot \left(-1 - m \cdot \frac{m}{v}\right)\]

Alternative 24
Error	10.5
Cost	576

\[\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\]

Alternative 25
Error	9.9
Cost	448

\[\left(m + \frac{m}{v}\right) + -1\]

Alternative 26
Error	37.0
Cost	64

\[-1\]

Alternative 27
Error	61.4
Cost	64

\[1\]

Alternative 28
Error	62.3
Cost	64

\[0\]

Error

Derivation

Initial program 0.1
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
Taylor expanded around 0 0.1
\[\leadsto \color{blue}{\left(m + \left(\frac{m}{v} + \frac{{m}^{3}}{v}\right)\right) - \left(2 \cdot \frac{{m}^{2}}{v} + 1\right)}\]
Simplified0.1
\[\leadsto \color{blue}{\left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right) + \frac{{m}^{3}}{v}}\]
Simplified0.1
\[\leadsto \color{blue}{\frac{{m}^{3}}{v} + \left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right)}\]
Final simplification0.1
\[\leadsto \frac{{m}^{3}}{v} + \left(\left(m + \frac{m}{v}\right) - \left(1 + 2 \cdot \left(m \cdot \frac{m}{v}\right)\right)\right)\]

Reproduce

herbie shell --seed 2021022 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))