b parameter of renormalized beta distribution

Average Accuracy: 99.9% → 99.1%

Time: 7.9s

Precision: binary64

Cost: 836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-36}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m \cdot \left(m + -2\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 9.2e-36)
   (+ m (+ -1.0 (/ m v)))
   (* (/ m v) (+ 1.0 (* m (+ m -2.0))))))

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 (m <= 9.2e-36) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m / v) * (1.0 + (m * (m + -2.0)));
	}
	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 (m <= 9.2d-36) then
        tmp = m + ((-1.0d0) + (m / v))
    else
        tmp = (m / v) * (1.0d0 + (m * (m + (-2.0d0))))
    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 (m <= 9.2e-36) {
		tmp = m + (-1.0 + (m / v));
	} else {
		tmp = (m / v) * (1.0 + (m * (m + -2.0)));
	}
	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 m <= 9.2e-36:
		tmp = m + (-1.0 + (m / v))
	else:
		tmp = (m / v) * (1.0 + (m * (m + -2.0)))
	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 (m <= 9.2e-36)
		tmp = Float64(m + Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m / v) * Float64(1.0 + Float64(m * Float64(m + -2.0))));
	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 (m <= 9.2e-36)
		tmp = m + (-1.0 + (m / v));
	else
		tmp = (m / v) * (1.0 + (m * (m + -2.0)));
	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[m, 9.2e-36], N[(m + N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * N[(1.0 + N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.19999999999999986e-36

   1. Initial program 100.0%

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

   2. Simplified 99.8%

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

      [Start] 100.0	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
      *-commutative [=>] 100.0	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      associate-*r/ [<=] 99.8	\[ \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} - 1\right) \]
      fma-neg [=>] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
      metadata-eval [=>] 99.8	\[ \left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 99.8%

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

   4. Simplified 100.0%

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

      [Start] 99.8	\[ \left(1 + \frac{1}{v}\right) \cdot m - 1 \]
      sub-neg [=>] 99.8	\[ \color{blue}{\left(1 + \frac{1}{v}\right) \cdot m + \left(-1\right)} \]
      metadata-eval [=>] 99.8	\[ \left(1 + \frac{1}{v}\right) \cdot m + \color{blue}{-1} \]
      +-commutative [=>] 99.8	\[ \color{blue}{-1 + \left(1 + \frac{1}{v}\right) \cdot m} \]
      +-commutative [=>] 99.8	\[ -1 + \color{blue}{\left(\frac{1}{v} + 1\right)} \cdot m \]
      distribute-lft1-in [<=] 99.8	\[ -1 + \color{blue}{\left(\frac{1}{v} \cdot m + m\right)} \]
      associate-*l/ [=>] 100.0	\[ -1 + \left(\color{blue}{\frac{1 \cdot m}{v}} + m\right) \]
      *-lft-identity [=>] 100.0	\[ -1 + \left(\frac{\color{blue}{m}}{v} + m\right) \]
      associate-+r+ [=>] 100.0	\[ \color{blue}{\left(-1 + \frac{m}{v}\right) + m} \]

   if 9.19999999999999986e-36 < m

   1. Initial program 99.5%

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

   2. Simplified 99.5%

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

      [Start] 99.5	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
      *-commutative [=>] 99.5	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 99.5	\[ \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      associate-*l/ [<=] 99.5	\[ \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 99.5	\[ \left(1 - m\right) \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

   3. Taylor expanded in v around 0 96.2%

      \[\leadsto \color{blue}{\frac{m \cdot {\left(1 - m\right)}^{2}}{v}} \]

   4. Simplified 96.2%

      \[\leadsto \color{blue}{{\left(1 - m\right)}^{2} \cdot \frac{m}{v}} \]
      Proof

      [Start] 96.2	\[ \frac{m \cdot {\left(1 - m\right)}^{2}}{v} \]
      associate-*l/ [<=] 96.2	\[ \color{blue}{\frac{m}{v} \cdot {\left(1 - m\right)}^{2}} \]
      *-commutative [=>] 96.2	\[ \color{blue}{{\left(1 - m\right)}^{2} \cdot \frac{m}{v}} \]

   5. Taylor expanded in m around 0 96.2%

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

   6. Simplified 96.2%

      \[\leadsto \color{blue}{\left(1 + m \cdot \left(m + -2\right)\right)} \cdot \frac{m}{v} \]
      Proof

      [Start] 96.2	\[ \left(1 + \left(-2 \cdot m + {m}^{2}\right)\right) \cdot \frac{m}{v} \]
      +-commutative [=>] 96.2	\[ \left(1 + \color{blue}{\left({m}^{2} + -2 \cdot m\right)}\right) \cdot \frac{m}{v} \]
      unpow2 [=>] 96.2	\[ \left(1 + \left(\color{blue}{m \cdot m} + -2 \cdot m\right)\right) \cdot \frac{m}{v} \]
      distribute-rgt-out [=>] 96.2	\[ \left(1 + \color{blue}{m \cdot \left(m + -2\right)}\right) \cdot \frac{m}{v} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.2 \cdot 10^{-36}:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m \cdot \left(m + -2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.6%
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{-161}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{-136}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	832

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

Alternative 3
Accuracy	99.9%
Cost	832

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

Alternative 4
Accuracy	99.9%
Cost	832

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

Alternative 5
Accuracy	97.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(m + -2\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	97.3%
Cost	708

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

Alternative 7
Accuracy	60.9%
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 1.18 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.38 \cdot 10^{-160}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 2.45 \cdot 10^{-138}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 8
Accuracy	96.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{m}{v}\\ \end{array} \]

Alternative 9
Accuracy	96.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;m + \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 10
Accuracy	42.1%
Cost	192

\[m + -1 \]

Alternative 11
Accuracy	41.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023151 
(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)))