b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 3.9s

Precision: binary64

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

Math

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

↓

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

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

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.1

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

2. Applied sub-neg_binary64 0.1

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

3. Applied distribute-rgt-in_binary64 0.1

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

4. Simplified 0.1

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

5. Simplified 0.1

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

6. Taylor expanded in m around 0 0.1

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

7. Simplified 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022131 
(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)))