b parameter of renormalized beta distribution

Average Accuracy: 58.4% → 60.3%

Time: 8.5s

Precision: binary64

Cost: 1732

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m - m \cdot m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= (* (- 1.0 m) (+ (/ (* m (- 1.0 m)) v) -1.0)) 2e+297)
   (* (- 1.0 m) (+ (/ (- m (* m m)) v) -1.0))
   (+ m -1.0)))

C

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

↓

double code(double m, double v) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= 2e+297) {
		tmp = (1.0 - m) * (((m - (m * m)) / v) + -1.0);
	} else {
		tmp = m + -1.0;
	}
	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) * (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 (((1.0d0 - m) * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= 2d+297) then
        tmp = (1.0d0 - m) * (((m - (m * m)) / v) + (-1.0d0))
    else
        tmp = m + (-1.0d0)
    end if
    code = tmp
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) {
	double tmp;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= 2e+297) {
		tmp = (1.0 - m) * (((m - (m * m)) / v) + -1.0);
	} else {
		tmp = m + -1.0;
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if ((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= 2e+297:
		tmp = (1.0 - m) * (((m - (m * m)) / v) + -1.0)
	else:
		tmp = m + -1.0
	return tmp

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)
	tmp = 0.0
	if (Float64(Float64(1.0 - m) * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= 2e+297)
		tmp = Float64(Float64(1.0 - m) * Float64(Float64(Float64(m - Float64(m * m)) / v) + -1.0));
	else
		tmp = Float64(m + -1.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (((1.0 - m) * (((m * (1.0 - m)) / v) + -1.0)) <= 2e+297)
		tmp = (1.0 - m) * (((m - (m * m)) / v) + -1.0);
	else
		tmp = m + -1.0;
	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] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

↓

code[m_, v_] := If[LessEqual[N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(1.0 - m), $MachinePrecision] * N[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) (-.f64 1 m)) < 2e297

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.9	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
      sub-neg [=>] 99.9	\[ \left(\frac{m \cdot \color{blue}{\left(1 + \left(-m\right)\right)}}{v} - 1\right) \cdot \left(1 - m\right) \]
      distribute-rgt-in [=>] 99.9	\[ \left(\frac{\color{blue}{1 \cdot m + \left(-m\right) \cdot m}}{v} - 1\right) \cdot \left(1 - m\right) \]
      *-un-lft-identity [<=] 99.9	\[ \left(\frac{\color{blue}{m} + \left(-m\right) \cdot m}{v} - 1\right) \cdot \left(1 - m\right) \]

   if 2e297 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) (-.f64 1 m))

   1. Initial program 0.0%

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

   2. Taylor expanded in v around inf 5.4%

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

   3. Simplified 5.4%

      \[\leadsto \color{blue}{m + -1} \]
      Step-by-step derivation

      [Start] 5.4	\[ -1 \cdot \left(1 - m\right) \]
      mul-1-neg [=>] 5.4	\[ \color{blue}{-\left(1 - m\right)} \]
      sub-neg [=>] 5.4	\[ -\color{blue}{\left(1 + \left(-m\right)\right)} \]
      +-commutative [<=] 5.4	\[ -\color{blue}{\left(\left(-m\right) + 1\right)} \]
      distribute-neg-in [=>] 5.4	\[ \color{blue}{\left(-\left(-m\right)\right) + \left(-1\right)} \]
      remove-double-neg [=>] 5.4	\[ \color{blue}{m} + \left(-1\right) \]
      metadata-eval [=>] 5.4	\[ m + \color{blue}{-1} \]

3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m - m \cdot m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	60.3%
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{\frac{v}{1 - m}}\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 2
Accuracy	59.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq 1.12 \cdot 10^{-14}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 3
Accuracy	59.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{1 - m}}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 4
Accuracy	59.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-26}:\\ \;\;\;\;\left(m + -1\right) + \frac{m}{v}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\frac{m + \left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 5
Accuracy	59.9%
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq 2.7 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 6
Accuracy	58.1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(m + -1\right) + \frac{m}{v}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\left(m + -1\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 7
Accuracy	58.1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\left(m + -1\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 8
Accuracy	58.1%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\left(m + -1\right) \cdot \frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 9
Accuracy	58.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 1.6:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 10
Accuracy	37.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 2.15 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m + \frac{m}{v}\\ \end{array} \]

Alternative 11
Accuracy	51.4%
Cost	448

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

Alternative 12
Accuracy	37.2%
Cost	324

\[\begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 13
Accuracy	27.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \]

Alternative 14
Accuracy	27.5%
Cost	192

\[m + -1 \]

Alternative 15
Accuracy	25.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023157 
(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)))