a parameter of renormalized beta distribution

Average Error: 0.2 → 0.3

Time: 41.6s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\frac{m}{v} - 0.5\right) + -0.5\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.8e-17)
   (* (+ (- (/ m v) 0.5) -0.5) m)
   (* m (* m (/ (- 1.0 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 <= 1.8e-17) {
		tmp = (((m / v) - 0.5) + -0.5) * m;
	} else {
		tmp = m * (m * ((1.0 - 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 <= 1.8d-17) then
        tmp = (((m / v) - 0.5d0) + (-0.5d0)) * m
    else
        tmp = m * (m * ((1.0d0 - 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 <= 1.8e-17) {
		tmp = (((m / v) - 0.5) + -0.5) * m;
	} else {
		tmp = m * (m * ((1.0 - 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 <= 1.8e-17:
		tmp = (((m / v) - 0.5) + -0.5) * m
	else:
		tmp = m * (m * ((1.0 - 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 <= 1.8e-17)
		tmp = Float64(Float64(Float64(Float64(m / v) - 0.5) + -0.5) * m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - 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 <= 1.8e-17)
		tmp = (((m / v) - 0.5) + -0.5) * m;
	else
		tmp = m * (m * ((1.0 - 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, 1.8e-17], N[(N[(N[(N[(m / v), $MachinePrecision] - 0.5), $MachinePrecision] + -0.5), $MachinePrecision] * m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.79999999999999997e-17

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\left(\left(\frac{m \cdot \left(1 - m\right)}{v} - 0.5\right) + -0.5\right)} \cdot m \]

   3. Taylor expanded in m around 0 0.1

      \[\leadsto \left(\left(\color{blue}{\frac{m}{v}} - 0.5\right) + -0.5\right) \cdot m \]

   if 1.79999999999999997e-17 < m

   1. Initial program 0.4

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

   2. Taylor expanded in v around 0 1.0

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

   3. Applied egg-rr 1.0

      \[\leadsto \color{blue}{m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right) + 0} \]

   4. Simplified 1.0

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

      [Start] 1.0	\[ m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right) + 0 \]
      rational_best-simplify-3 [=>] 1.0	\[ \color{blue}{0 + m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)} \]
      rational_best-simplify-6 [=>] 1.0	\[ \color{blue}{m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)} \]
      rational_best-simplify-55 [=>] 1.0	\[ m \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\frac{m}{v} - 0.5\right) + -0.5\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	832

\[\left(\left(\frac{m \cdot \left(1 - m\right)}{v} - 0.5\right) + -0.5\right) \cdot m \]

Alternative 2
Error	24.3
Cost	716

\[\begin{array}{l} t_0 := \frac{m}{v} \cdot m\\ \mathbf{if}\;v \leq 1.7 \cdot 10^{-210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-199}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 9.8 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	0.2
Cost	704

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

Alternative 6
Error	10.1
Cost	448

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

Alternative 7
Error	36.3
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023099 
(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))