a parameter of renormalized beta distribution

Average Error: 0.2 → 0.4

Time: 6.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.3018175972005351 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.3018175972005351e-19)
   (* m (+ (/ m v) -1.0))
   (* (- 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 <= 1.3018175972005351e-19) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 1.3018175972005351d-19) then
        tmp = m * ((m / v) + (-1.0d0))
    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 <= 1.3018175972005351e-19) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 1.3018175972005351e-19:
		tmp = m * ((m / v) + -1.0)
	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 <= 1.3018175972005351e-19)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	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 <= 1.3018175972005351e-19)
		tmp = m * ((m / v) + -1.0);
	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, 1.3018175972005351e-19], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $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 < 1.30181759720053515e-19

   1. Initial program 0.1

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

   2. Taylor expanded in m around 0 0.1

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

   if 1.30181759720053515e-19 < m

   1. Initial program 0.4

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

   2. Taylor expanded in m around inf 1.2

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

   3. Simplified 1.3

      \[\leadsto \color{blue}{\frac{m}{\frac{v}{m}} \cdot \left(1 - m\right)} \]
      Proof
      (*.f64 (/.f64 m (/.f64 v m)) (-.f64 1 m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l* (/.f64 (*.f64 m m) v)) (-.f64 1 m)): 45 points increase in error, 60 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2 (pow.f64 m 2)) v) (-.f64 1 m)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out-- (-.f64 (*.f64 1 (/.f64 (pow.f64 m 2) v)) (*.f64 m (/.f64 (pow.f64 m 2) v)))): 2 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> *-lft-identity (/.f64 (pow.f64 m 2) v)) (*.f64 m (/.f64 (pow.f64 m 2) v))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (pow.f64 m 2) v) (Rewrite<= *-commutative (*.f64 (/.f64 (pow.f64 m 2) v) m))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (pow.f64 m 2) v) (Rewrite=> associate-*l/ (/.f64 (*.f64 (pow.f64 m 2) m) v))): 9 points increase in error, 7 points decrease in error
      (-.f64 (/.f64 (pow.f64 m 2) v) (/.f64 (*.f64 (Rewrite=> unpow2 (*.f64 m m)) m) v)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (pow.f64 m 2) v) (/.f64 (Rewrite<= unpow3 (pow.f64 m 3)) v)): 2 points increase in error, 10 points decrease in error
      (Rewrite<= unsub-neg (+.f64 (/.f64 (pow.f64 m 2) v) (neg.f64 (/.f64 (pow.f64 m 3) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 m 2) v) (Rewrite<= mul-1-neg (*.f64 -1 (/.f64 (pow.f64 m 3) v)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative (+.f64 (*.f64 -1 (/.f64 (pow.f64 m 3) v)) (/.f64 (pow.f64 m 2) v))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.3018175972005351 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.3018175972005351 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

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

Alternative 3
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{-m}}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{-v}\\ \end{array} \]

Alternative 5
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(-m\right)\right)\\ \end{array} \]

Alternative 6
Error	2.4
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{-v}{m \cdot m}}\\ \end{array} \]

Alternative 7
Error	24.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 2.5597289303195304 \cdot 10^{-180}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Error	24.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 2.5597289303195304 \cdot 10^{-180}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9
Error	10.3
Cost	448

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

Alternative 10
Error	10.3
Cost	448

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

Alternative 11
Error	36.9
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022318 
(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))