b parameter of renormalized beta distribution

Average Accuracy: 99.9% → 99.9%

Time: 8.0s

Precision: binary64

Cost: 1472

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (+ (/ m v) (- (- -1.0 (* m (/ m v))) (* m (+ -1.0 (/ m (/ v (- 1.0 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 / v) + ((-1.0 - (m * (m / v))) - (m * (-1.0 + (m / (v / (1.0 - 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 / v) + (((-1.0d0) - (m * (m / v))) - (m * ((-1.0d0) + (m / (v / (1.0d0 - 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 / v) + ((-1.0 - (m * (m / v))) - (m * (-1.0 + (m / (v / (1.0 - m))))));
}

Python

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

↓

def code(m, v):
	return (m / v) + ((-1.0 - (m * (m / v))) - (m * (-1.0 + (m / (v / (1.0 - 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 / v) + Float64(Float64(-1.0 - Float64(m * Float64(m / v))) - Float64(m * Float64(-1.0 + Float64(m / Float64(v / Float64(1.0 - 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 / v) + ((-1.0 - (m * (m / v))) - (m * (-1.0 + (m / (v / (1.0 - 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 / v), $MachinePrecision] + N[(N[(-1.0 - N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(m * N[(-1.0 + N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in m around 0 99.9%

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

3. Simplified 99.9%

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

   [Start] 99.9	\[ \left(\frac{-1 \cdot {m}^{2} + m}{v} - 1\right) \cdot \left(1 - m\right) \]
   +-commutative [=>] 99.9	\[ \left(\frac{\color{blue}{m + -1 \cdot {m}^{2}}}{v} - 1\right) \cdot \left(1 - m\right) \]
   mul-1-neg [=>] 99.9	\[ \left(\frac{m + \color{blue}{\left(-{m}^{2}\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]
   unpow2 [=>] 99.9	\[ \left(\frac{m + \left(-\color{blue}{m \cdot m}\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   sub-neg [<=] 99.9	\[ \left(\frac{\color{blue}{m - m \cdot m}}{v} - 1\right) \cdot \left(1 - m\right) \]

4. Applied egg-rr 99.9%

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

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

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\left(m + -1\right) \cdot \left(1 - \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(\left(m + -1\right) \cdot \left(m + -1\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\left(m + -1\right) \cdot \left(1 - \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 - m \cdot \left(2 - m\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{m}{v} + -1}{\frac{1}{1 - m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 - m \cdot \left(2 - m\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{m}{v} + -1}{\frac{1}{1 - m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	832

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

Alternative 6
Accuracy	99.9%
Cost	832

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

Alternative 7
Accuracy	99.9%
Cost	832

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

Alternative 8
Accuracy	96.3%
Cost	708

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

Alternative 9
Accuracy	96.4%
Cost	708

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

Alternative 10
Accuracy	97.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.65:\\ \;\;\;\;\left(m + -1\right) \cdot \left(1 - \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	96.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.7:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 12
Accuracy	61.6%
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 13
Accuracy	85.2%
Cost	320

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

Alternative 14
Accuracy	42.6%
Cost	192

\[m + -1 \]

Alternative 15
Accuracy	42.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023135 
(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)))