a parameter of renormalized beta distribution

Average Error: 0.2 → 0.4

Time: 5.0s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.7020584006126057 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.7020584006126057e-24)
   (- (* m (/ m v)) m)
   (* (- 1.0 m) (/ m (/ v m)))))

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 <= 1.7020584006126057e-24) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (1.0 - m) * (m / (v / m));
	}
	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 <= 1.7020584006126057d-24) then
        tmp = (m * (m / v)) - m
    else
        tmp = (1.0d0 - m) * (m / (v / m))
    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 <= 1.7020584006126057e-24) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = (1.0 - m) * (m / (v / m));
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if m <= 1.7020584006126057e-24:
		tmp = (m * (m / v)) - m
	else:
		tmp = (1.0 - m) * (m / (v / m))
	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 <= 1.7020584006126057e-24)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m / Float64(v / m)));
	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 <= 1.7020584006126057e-24)
		tmp = (m * (m / v)) - m;
	else
		tmp = (1.0 - m) * (m / (v / m));
	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, 1.7020584006126057e-24], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.70205840061260573e-24

   1. Initial program 0.1

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

   2. Applied egg-rr 0.3

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

   3. Taylor expanded in m around 0 9.0

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

   4. Simplified 9.0

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

   5. Taylor expanded in m around 0 9.0

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

   6. Simplified 0.1

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

   if 1.70205840061260573e-24 < m

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   3. Taylor expanded in v around 0 1.3

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

   4. Applied egg-rr 1.3

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

   5. Applied egg-rr 1.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.7020584006126057 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.0
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 3.303308531097567 \cdot 10^{-181}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;v \leq 8.239454599011323 \cdot 10^{-126}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 3.5622284953776748 \cdot 10^{-118}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	24.0
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 3.303308531097567 \cdot 10^{-181}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;v \leq 8.239454599011323 \cdot 10^{-126}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 3.5622284953776748 \cdot 10^{-118}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	23.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 3.303308531097567 \cdot 10^{-181}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}}\\ \mathbf{elif}\;v \leq 8.239454599011323 \cdot 10^{-126}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 3.5622284953776748 \cdot 10^{-118}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	0.2
Cost	704

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

Alternative 6
Error	0.2
Cost	704

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

Alternative 7
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.0028734812165592764:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(-m\right)}}\\ \end{array} \]

Alternative 8
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.0028734812165592764:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{-m}{v}\right)\\ \end{array} \]

Alternative 9
Error	25.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 6.598331315163483 \cdot 10^{-132}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \]

Alternative 10
Error	10.5
Cost	448

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

Alternative 11
Error	36.7
Cost	128

\[-m \]

Alternative 12
Error	60.8
Cost	64

\[m \]

Reproduce

herbie shell --seed 2022228 
(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))