Average Error: 0.1 → 0.1

Time: 2.0min

Precision: binary64

Cost: 1344

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

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

↓

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

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

↓

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

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

↓

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

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

↓

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

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	0.1
Cost	832

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

Alternative 2
Error	0.4
Cost	1025

\[\begin{array}{l} \mathbf{if}\;m \leq 8.58952850843879 \cdot 10^{-28}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array}\]

Alternative 3
Error	0.4
Cost	1025

\[\begin{array}{l} \mathbf{if}\;m \leq 8.58952850843879 \cdot 10^{-28}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m + \left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array}\]

Alternative 4
Error	2.5
Cost	897

\[\begin{array}{l} \mathbf{if}\;m \leq 0.9958808451382868:\\ \;\;\;\;-1 + \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(1 + m \cdot \frac{m}{v}\right)\\ \end{array}\]

Alternative 5
Error	9.8
Cost	448

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

Alternative 6
Error	9.8
Cost	320

\[\frac{m}{v} + -1\]

Alternative 7
Error	36.9
Cost	192

\[m + -1\]

Alternative 8
Error	37.2
Cost	64

\[-1\]

Derivation

Initial program 0.1
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
Using strategy rm
Applied sub-neg_binary640.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)}\]
Applied distribute-rgt-in_binary640.1
\[\leadsto \color{blue}{1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(-m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)}\]
Simplified0.1
\[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} + \left(-m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)\]
Simplified0.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)}\]
Using strategy rm
Applied associate-/l*_binary640.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} - 1\right) \cdot \left(-m\right)\]
Using strategy rm
Applied distribute-rgt-neg-out_binary640.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \color{blue}{\left(-\left(\frac{m}{\frac{v}{1 - m}} - 1\right) \cdot m\right)}\]
Simplified0.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \left(-\color{blue}{m \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right)}\right)\]
Simplified0.1
\[\leadsto \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) - m \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right)}\]
Final simplification0.1
\[\leadsto \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) - m \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right)\]

Reproduce

herbie shell --seed 2021065 
(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)))