b parameter of renormalized beta distribution

Average Accuracy: 99.9% → 99.9%

Time: 10.0s

Precision: binary64

Cost: 1344

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

↓

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

FPCore

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

↓

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

C

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 + ((m / v) * (1.0 - m))) + (-1.0 - ((1.0 - m) * (m / (v / m))));
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

↓

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (m + ((m / v) * (1.0d0 - m))) + ((-1.0d0) - ((1.0d0 - m) * (m / (v / m))))
end function

Java

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

↓

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

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

↓

def code(m, v):
	return (m + ((m / v) * (1.0 - m))) + (-1.0 - ((1.0 - m) * (m / (v / m))))

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

↓

function code(m, v)
	return Float64(Float64(m + Float64(Float64(m / v) * Float64(1.0 - m))) + Float64(-1.0 - Float64(Float64(1.0 - m) * Float64(m / Float64(v / m)))))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

↓

function tmp = code(m, v)
	tmp = (m + ((m / v) * (1.0 - m))) + (-1.0 - ((1.0 - m) * (m / (v / m))));
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

↓

code[m_, v_] := N[(N[(m + N[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(N[(1.0 - m), $MachinePrecision] * N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Applied egg-rr 99.7%

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

   [Start] 99.9	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   sub-neg [=>] 99.9	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)} \]
   distribute-lft-in [=>] 99.9	\[ \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot 1 + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)} \]
   *-commutative [<=] 99.9	\[ \color{blue}{1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right) \]
   *-un-lft-identity [<=] 99.9	\[ \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} + \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right) \]
   associate-+l- [=>] 99.9	\[ \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} - \left(1 - \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)\right)} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{m}{\frac{v}{1 - m}}} - \left(1 - \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)\right) \]
   div-inv [=>] 99.7	\[ \color{blue}{m \cdot \frac{1}{\frac{v}{1 - m}}} - \left(1 - \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)\right) \]
   associate-/l* [<=] 99.7	\[ m \cdot \color{blue}{\frac{1 \cdot \left(1 - m\right)}{v}} - \left(1 - \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)\right) \]
   *-un-lft-identity [<=] 99.7	\[ m \cdot \frac{\color{blue}{1 - m}}{v} - \left(1 - \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(-m\right)\right) \]
   sub-neg [=>] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \cdot \left(-m\right)\right) \]
   associate-/l* [=>] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} + \left(-1\right)\right) \cdot \left(-m\right)\right) \]
   div-inv [=>] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \left(\color{blue}{m \cdot \frac{1}{\frac{v}{1 - m}}} + \left(-1\right)\right) \cdot \left(-m\right)\right) \]
   associate-/l* [<=] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \left(m \cdot \color{blue}{\frac{1 \cdot \left(1 - m\right)}{v}} + \left(-1\right)\right) \cdot \left(-m\right)\right) \]
   *-un-lft-identity [<=] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \left(m \cdot \frac{\color{blue}{1 - m}}{v} + \left(-1\right)\right) \cdot \left(-m\right)\right) \]
   metadata-eval [=>] 99.7	\[ m \cdot \frac{1 - m}{v} - \left(1 - \left(m \cdot \frac{1 - m}{v} + \color{blue}{-1}\right) \cdot \left(-m\right)\right) \]

3. Taylor expanded in v around 0 99.9%

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

4. Simplified 99.9%

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

   [Start] 99.9	\[ \left(-1 \cdot \frac{{m}^{2} \cdot \left(1 - m\right)}{v} + \left(m + \frac{m \cdot \left(1 - m\right)}{v}\right)\right) - 1 \]
   sub-neg [=>] 99.9	\[ \color{blue}{\left(-1 \cdot \frac{{m}^{2} \cdot \left(1 - m\right)}{v} + \left(m + \frac{m \cdot \left(1 - m\right)}{v}\right)\right) + \left(-1\right)} \]
   +-commutative [<=] 99.9	\[ \left(-1 \cdot \frac{{m}^{2} \cdot \left(1 - m\right)}{v} + \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right)}\right) + \left(-1\right) \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right) + -1 \cdot \frac{{m}^{2} \cdot \left(1 - m\right)}{v}\right)} + \left(-1\right) \]
   mul-1-neg [=>] 99.9	\[ \left(\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right) + \color{blue}{\left(-\frac{{m}^{2} \cdot \left(1 - m\right)}{v}\right)}\right) + \left(-1\right) \]
   unsub-neg [=>] 99.9	\[ \color{blue}{\left(\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right) - \frac{{m}^{2} \cdot \left(1 - m\right)}{v}\right)} + \left(-1\right) \]
   metadata-eval [=>] 99.9	\[ \left(\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right) - \frac{{m}^{2} \cdot \left(1 - m\right)}{v}\right) + \color{blue}{-1} \]
   associate-+l- [=>] 99.9	\[ \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + m\right) - \left(\frac{{m}^{2} \cdot \left(1 - m\right)}{v} - -1\right)} \]
   +-commutative [=>] 99.9	\[ \color{blue}{\left(m + \frac{m \cdot \left(1 - m\right)}{v}\right)} - \left(\frac{{m}^{2} \cdot \left(1 - m\right)}{v} - -1\right) \]
   associate-/l* [=>] 99.9	\[ \left(m + \color{blue}{\frac{m}{\frac{v}{1 - m}}}\right) - \left(\frac{{m}^{2} \cdot \left(1 - m\right)}{v} - -1\right) \]
   associate-/r/ [=>] 99.9	\[ \left(m + \color{blue}{\frac{m}{v} \cdot \left(1 - m\right)}\right) - \left(\frac{{m}^{2} \cdot \left(1 - m\right)}{v} - -1\right) \]
   associate-/l* [=>] 99.9	\[ \left(m + \frac{m}{v} \cdot \left(1 - m\right)\right) - \left(\color{blue}{\frac{{m}^{2}}{\frac{v}{1 - m}}} - -1\right) \]
   associate-/r/ [=>] 99.9	\[ \left(m + \frac{m}{v} \cdot \left(1 - m\right)\right) - \left(\color{blue}{\frac{{m}^{2}}{v} \cdot \left(1 - m\right)} - -1\right) \]
   unpow2 [=>] 99.9	\[ \left(m + \frac{m}{v} \cdot \left(1 - m\right)\right) - \left(\frac{\color{blue}{m \cdot m}}{v} \cdot \left(1 - m\right) - -1\right) \]
   associate-/l* [=>] 99.9	\[ \left(m + \frac{m}{v} \cdot \left(1 - m\right)\right) - \left(\color{blue}{\frac{m}{\frac{v}{m}}} \cdot \left(1 - m\right) - -1\right) \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	72.4%
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.45 \cdot 10^{-118}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{-107}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 2
Accuracy	72.2%
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 4.2 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-118}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.7 \cdot 10^{-107}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{1 - m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 3
Accuracy	72.3%
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-161}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.6 \cdot 10^{-118}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{-107}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 10^{-23}:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m - m \cdot m}}\\ \end{array} \]

Alternative 6
Accuracy	99.9%
Cost	832

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

Alternative 7
Accuracy	96.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(m + -1\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	96.2%
Cost	708

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

Alternative 9
Accuracy	96.2%
Cost	708

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

Alternative 10
Accuracy	96.2%
Cost	708

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

Alternative 11
Accuracy	96.2%
Cost	708

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

Alternative 12
Accuracy	61.4%
Cost	589

\[\begin{array}{l} \mathbf{if}\;v \leq 2.75 \cdot 10^{-161} \lor \neg \left(v \leq 2.3 \cdot 10^{-139}\right) \land v \leq 1.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 13
Accuracy	96.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 14
Accuracy	41.7%
Cost	192

\[m + -1 \]

Alternative 15
Accuracy	41.2%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023131 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))