b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 8.2s

Precision: binary64

Cost: 7296

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

Math

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

↓

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

FPCore

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

↓

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

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 (1.0 - m) * ((m / v) + (-1.0 - (pow(m, 2.0) / v)));
}

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 = (1.0d0 - m) * ((m / v) + ((-1.0d0) - ((m ** 2.0d0) / v)))
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 (1.0 - m) * ((m / v) + (-1.0 - (Math.pow(m, 2.0) / v)));
}

Python

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

↓

def code(m, v):
	return (1.0 - m) * ((m / v) + (-1.0 - (math.pow(m, 2.0) / v)))

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(1.0 - m) * Float64(Float64(m / v) + Float64(-1.0 - Float64((m ^ 2.0) / v))))
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 = (1.0 - m) * ((m / v) + (-1.0 - ((m ^ 2.0) / v)));
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[(1.0 - m), $MachinePrecision] * N[(N[(m / v), $MachinePrecision] + N[(-1.0 - N[(N[Power[m, 2.0], $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   rational_best_45_simplify-91 [=>] 0.1	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
   rational_best_45_simplify-56 [=>] 0.1	\[ \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right)} \]
   rational_best_45_simplify-67 [=>] 0.1	\[ \left(1 - m\right) \cdot \left(\frac{\color{blue}{1 \cdot m - m \cdot m}}{v} + -1\right) \]
   rational_best_45_simplify-91 [=>] 0.1	\[ \left(1 - m\right) \cdot \left(\frac{\color{blue}{m \cdot 1} - m \cdot m}{v} + -1\right) \]
   rational_best_45_simplify-8 [=>] 0.1	\[ \left(1 - m\right) \cdot \left(\frac{\color{blue}{m} - m \cdot m}{v} + -1\right) \]

3. Taylor expanded in m around 0 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \left(1 - m\right) \cdot \left(\left(\frac{m}{v} + -1 \cdot \frac{{m}^{2}}{v}\right) + -1\right) \]
   rational_best_45_simplify-91 [=>] 0.1	\[ \left(1 - m\right) \cdot \left(\left(\frac{m}{v} + \color{blue}{\frac{{m}^{2}}{v} \cdot -1}\right) + -1\right) \]
   rational_best_45_simplify-16 [=>] 0.1	\[ \left(1 - m\right) \cdot \left(\left(\frac{m}{v} + \color{blue}{\left(-\frac{{m}^{2}}{v}\right)}\right) + -1\right) \]

5. Applied egg-rr 0.1

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

6. Simplified 0.1

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

   [Start] 0.1	\[ m + \left(-1 + \left(m + -1\right) \cdot \left(\frac{{m}^{2}}{v} - \frac{m}{v}\right)\right) \]
   rational_best_45_simplify-73 [=>] 0.1	\[ m + \color{blue}{\left(\left(m + -1\right) \cdot \left(\frac{{m}^{2}}{v} - \frac{m}{v}\right) + -1\right)} \]
   rational_best_45_simplify-80 [<=] 0.1	\[ \color{blue}{\left(m + -1\right) \cdot \left(\frac{{m}^{2}}{v} - \frac{m}{v}\right) + \left(m + -1\right)} \]
   rational_best_45_simplify-73 [=>] 0.1	\[ \color{blue}{\left(m + -1\right) + \left(m + -1\right) \cdot \left(\frac{{m}^{2}}{v} - \frac{m}{v}\right)} \]
   rational_best_45_simplify-99 [=>] 0.1	\[ \left(m + -1\right) + \color{blue}{\left(-\left(m + -1\right)\right) \cdot \left(\frac{m}{v} - \frac{{m}^{2}}{v}\right)} \]
   rational_best_45_simplify-2 [<=] 0.1	\[ \left(m + -1\right) + \color{blue}{\left(0 - \left(m + -1\right)\right)} \cdot \left(\frac{m}{v} - \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-67 [=>] 0.1	\[ \left(m + -1\right) + \color{blue}{\left(\frac{m}{v} \cdot \left(0 - \left(m + -1\right)\right) - \left(0 - \left(m + -1\right)\right) \cdot \frac{{m}^{2}}{v}\right)} \]
   rational_best_45_simplify-55 [=>] 0.1	\[ \left(m + -1\right) + \left(\frac{m}{v} \cdot \left(0 - \color{blue}{\left(m - 1\right)}\right) - \left(0 - \left(m + -1\right)\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-51 [=>] 0.1	\[ \left(m + -1\right) + \left(\frac{m}{v} \cdot \color{blue}{\left(1 - \left(m - 0\right)\right)} - \left(0 - \left(m + -1\right)\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-17 [=>] 0.1	\[ \left(m + -1\right) + \left(\frac{m}{v} \cdot \left(1 - \color{blue}{m}\right) - \left(0 - \left(m + -1\right)\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-91 [<=] 0.1	\[ \left(m + -1\right) + \left(\color{blue}{\left(1 - m\right) \cdot \frac{m}{v}} - \left(0 - \left(m + -1\right)\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-55 [=>] 0.1	\[ \left(m + -1\right) + \left(\left(1 - m\right) \cdot \frac{m}{v} - \left(0 - \color{blue}{\left(m - 1\right)}\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-51 [=>] 0.1	\[ \left(m + -1\right) + \left(\left(1 - m\right) \cdot \frac{m}{v} - \color{blue}{\left(1 - \left(m - 0\right)\right)} \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-17 [=>] 0.1	\[ \left(m + -1\right) + \left(\left(1 - m\right) \cdot \frac{m}{v} - \left(1 - \color{blue}{m}\right) \cdot \frac{{m}^{2}}{v}\right) \]
   rational_best_45_simplify-91 [<=] 0.1	\[ \left(m + -1\right) + \left(\left(1 - m\right) \cdot \frac{m}{v} - \color{blue}{\frac{{m}^{2}}{v} \cdot \left(1 - m\right)}\right) \]
   rational_best_45_simplify-109 [<=] 0.1	\[ \color{blue}{\left(\left(1 - m\right) \cdot \frac{m}{v} + \left(m + -1\right)\right) - \frac{{m}^{2}}{v} \cdot \left(1 - m\right)} \]

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	24.1
Cost	852

\[\begin{array}{l} \mathbf{if}\;v \leq 1.7 \cdot 10^{-213}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-203}:\\ \;\;\;\;m + -1\\ \mathbf{elif}\;v \leq 1.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-131}:\\ \;\;\;\;m + -1\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 2
Error	0.1
Cost	832

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

Alternative 3
Error	9.6
Cost	448

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

Alternative 4
Error	9.6
Cost	320

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

Alternative 5
Error	36.7
Cost	192

\[m + -1 \]

Alternative 6
Error	37.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023098 
(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)))