a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 13

Speedup: 1.0×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = m * (((m * (1.0d0 - m)) / v) + (-1.0d0))
end function

Java

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

Python

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

Julia

function code(m, v)
	return Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \]
4. Add Preprocessing

Alternative 2: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{-v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= (* m (+ (/ (* m (- 1.0 m)) v) -1.0)) -1e+47)
   (/ (* m (* m m)) (- v))
   (- (* m (/ m v)) m)))

C

double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e+47) {
		tmp = (m * (m * m)) / -v;
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m * (((m * (1.0d0 - m)) / v) + (-1.0d0))) <= (-1d+47)) then
        tmp = (m * (m * m)) / -v
    else
        tmp = (m * (m / v)) - m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e+47) {
		tmp = (m * (m * m)) / -v;
	} else {
		tmp = (m * (m / v)) - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e+47:
		tmp = (m * (m * m)) / -v
	else:
		tmp = (m * (m / v)) - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0)) <= -1e+47)
		tmp = Float64(Float64(m * Float64(m * m)) / Float64(-v));
	else
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m * (((m * (1.0 - m)) / v) + -1.0)) <= -1e+47)
		tmp = (m * (m * m)) / -v;
	else
		tmp = (m * (m / v)) - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], -1e+47], N[(N[(m * N[(m * m), $MachinePrecision]), $MachinePrecision] / (-v)), $MachinePrecision], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1e47

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{v} \cdot {m}^{3} \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v}\right)\right)} \cdot {m}^{3} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{3}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{m}^{3} \cdot \frac{1}{v}}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({m}^{3} \cdot \frac{1}{v}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3} \cdot 1}{v}}\right) \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{m}^{3}}}{v}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{m}^{3}}{v}}\right) \]

      11. cube-mult N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -1e47 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto m \cdot \color{blue}{\frac{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3} - {1}^{3}}{\frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v} + \left(1 \cdot 1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot 1\right)}} \]

      5. clear-num N/A

         \[\leadsto m \cdot \color{blue}{\frac{1}{\frac{\frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v} + \left(1 \cdot 1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot 1\right)}{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3} - {1}^{3}}}} \]

      6. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{m}{\frac{\frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v} + \left(1 \cdot 1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot 1\right)}{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3} - {1}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{m}{\frac{\frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v} + \left(1 \cdot 1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot 1\right)}{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3} - {1}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{m}{\color{blue}{\frac{1}{\frac{{\left(\frac{m \cdot \left(1 - m\right)}{v}\right)}^{3} - {1}^{3}}{\frac{m \cdot \left(1 - m\right)}{v} \cdot \frac{m \cdot \left(1 - m\right)}{v} + \left(1 \cdot 1 + \frac{m \cdot \left(1 - m\right)}{v} \cdot 1\right)}}}} \]

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto {m}^{3} \cdot \color{blue}{\left(\frac{1}{m \cdot v} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right)\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\frac{1}{m \cdot v} \cdot {m}^{3} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3}} \]

      3. unpow3 N/A

         \[\leadsto \frac{1}{m \cdot v} \cdot \color{blue}{\left(\left(m \cdot m\right) \cdot m\right)} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{m \cdot v} \cdot \left(\color{blue}{{m}^{2}} \cdot m\right) + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{m \cdot v} \cdot {m}^{2}\right) \cdot m} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      6. associate-/r* N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{1}{m}}{v}} \cdot {m}^{2}\right) \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{m} \cdot {m}^{2}}{v}} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      8. unpow2 N/A

         \[\leadsto \frac{\frac{1}{m} \cdot \color{blue}{\left(m \cdot m\right)}}{v} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{m} \cdot m\right) \cdot m}}{v} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{\color{blue}{1} \cdot m}{v} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      11. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{m}}{v} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      12. *-lft-identity N/A

          \[\leadsto \frac{\color{blue}{1 \cdot m}}{v} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      13. associate-*l/ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{v} \cdot m\right)} \cdot m + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      14. associate-*r* N/A

          \[\leadsto \color{blue}{\frac{1}{v} \cdot \left(m \cdot m\right)} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{v} \cdot \color{blue}{{m}^{2}} + \left(\mathsf{neg}\left(\frac{1}{v}\right)\right) \cdot {m}^{3} \]

      16. distribute-lft-neg-in N/A

          \[\leadsto \frac{1}{v} \cdot {m}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{3}\right)\right)} \]

      17. cube-mult N/A

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

   7. Applied rewrites 41.1%

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

   8. Taylor expanded in m around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. associate-/l* N/A

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

      5. unpow2 N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \frac{{m}^{2}}{v} + \color{blue}{\left(\mathsf{neg}\left(m\right)\right)} \]

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 97.3

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

   10. Applied rewrites 97.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} + -1\right) \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot m\right)}{-v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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))