a parameter of renormalized beta distribution

Average Error: 0.2 → 0.3

Time: 5.0s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 2.916119888016924 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 2.916119888016924e-16)
   (- (* m (/ m v)) m)
   (/ (* (- 1.0 m) (* m m)) v)))

C

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

↓

double code(double m, double v) {
	double tmp;
	if (m <= 2.916119888016924e-16) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

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) * m
end function

↓

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.916119888016924d-16) then
        tmp = (m * (m / v)) - m
    else
        tmp = ((1.0d0 - m) * (m * m)) / v
    end if
    code = tmp
end function

Java

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

↓

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.916119888016924e-16) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if m <= 2.916119888016924e-16:
		tmp = (m * (m / v)) - m
	else:
		tmp = ((1.0 - m) * (m * m)) / v
	return tmp

Julia

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

↓

function code(m, v)
	tmp = 0.0
	if (m <= 2.916119888016924e-16)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.916119888016924e-16)
		tmp = (m * (m / v)) - m;
	else
		tmp = ((1.0 - m) * (m * m)) / v;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[m_, v_] := If[LessEqual[m, 2.916119888016924e-16], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.91611988801692385e-16

   1. Initial program 0.1

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

   2. Simplified 0.2

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

   3. Taylor expanded in m around 0 8.2

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{m \cdot \frac{m}{v} - m} \]

   if 2.91611988801692385e-16 < m

   1. Initial program 0.3

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

   2. Taylor expanded in m around inf 0.9

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

   3. Simplified 1.0

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

   4. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{\left(m \cdot m\right) \cdot \left(1 - m\right)}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.916119888016924 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.6
Cost	716

\[\begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;v \leq 2.655211190641574 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 4.591078059283095 \cdot 10^{-170}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 1.8127622673361344 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	24.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 2.655211190641574 \cdot 10^{-200}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;v \leq 4.591078059283095 \cdot 10^{-170}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 1.8127622673361344 \cdot 10^{-157}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.916119888016924 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	0.2
Cost	704

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

Alternative 6
Error	2.2
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.015393630613039362:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(-m\right)\right)}{v}\\ \end{array} \]

Alternative 7
Error	14.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;v \leq 2.6 \cdot 10^{-234}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \end{array} \]

Alternative 8
Error	10.4
Cost	448

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

Alternative 9
Error	36.3
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022284 
(FPCore (m v)
  :name "a 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) m))