a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 14

Speedup: 1.0×

• 41.9% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{-85}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.04e-85) (* m (+ -1.0 (/ m v))) (/ (* m (- m (fma m m v))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.04e-85) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (m * (m - fma(m, m, v))) / v;
	}
	return tmp;
}

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.04e-85)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(m - fma(m, m, v))) / v);
	end
	return tmp
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.04e-85], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m - N[(m * m + v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.04e-85

   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 0

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

      1. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.04e-85 < 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 99.8%

      \[\leadsto \color{blue}{\frac{m}{\frac{1}{\mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in m around 0

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

   9. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{m \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.04 \cdot 10^{-85}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m - \mathsf{fma}\left(m, m, v\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{m \cdot m}{m}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ -1.0 (/ (* m (- 1.0 m)) v)))))
   (if (<= t_0 -1e+104)
     (- (/ (* m m) m))
     (if (<= t_0 -2e-308) (- (/ (* m m) v) m) (* m (/ m v))))))

C

double code(double m, double v) {
	double t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	double tmp;
	if (t_0 <= -1e+104) {
		tmp = -((m * m) / m);
	} else if (t_0 <= -2e-308) {
		tmp = ((m * m) / v) - m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * ((-1.0d0) + ((m * (1.0d0 - m)) / v))
    if (t_0 <= (-1d+104)) then
        tmp = -((m * m) / m)
    else if (t_0 <= (-2d-308)) then
        tmp = ((m * m) / v) - m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	double tmp;
	if (t_0 <= -1e+104) {
		tmp = -((m * m) / m);
	} else if (t_0 <= -2e-308) {
		tmp = ((m * m) / v) - m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m * (-1.0 + ((m * (1.0 - m)) / v))
	tmp = 0
	if t_0 <= -1e+104:
		tmp = -((m * m) / m)
	elif t_0 <= -2e-308:
		tmp = ((m * m) / v) - m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m * Float64(-1.0 + Float64(Float64(m * Float64(1.0 - m)) / v)))
	tmp = 0.0
	if (t_0 <= -1e+104)
		tmp = Float64(-Float64(Float64(m * m) / m));
	elseif (t_0 <= -2e-308)
		tmp = Float64(Float64(Float64(m * m) / v) - m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m * (-1.0 + ((m * (1.0 - m)) / v));
	tmp = 0.0;
	if (t_0 <= -1e+104)
		tmp = -((m * m) / m);
	elseif (t_0 <= -2e-308)
		tmp = ((m * m) / v) - m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m * N[(-1.0 + N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+104], (-N[(N[(m * m), $MachinePrecision] / m), $MachinePrecision]), If[LessEqual[t$95$0, -2e-308], N[(N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1e104

   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 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 5.6

         \[\leadsto \color{blue}{-m} \]

   5. Applied rewrites 5.6%

      \[\leadsto \color{blue}{-m} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.6%

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

   if -1e104 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   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 0

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

      1. distribute-lft-out-- N/A

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

      2. associate-/l* N/A

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

      3. unpow2 N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m)

   1. Initial program 99.6%

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

   3. Taylor expanded in m around inf

      \[\leadsto \color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)} \cdot m \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. rgt-mult-inverse N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. distribute-lft-out-- N/A

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

      12. div-sub N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

      15. distribute-rgt-out-- N/A

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

      16. *-lft-identity N/A

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

      17. unpow2 N/A

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

      18. lower--.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in m around 0

      \[\leadsto \frac{m}{\color{blue}{v}} \cdot m \]
   7. Step-by-step derivation

   8. Applied rewrites 92.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -1 \cdot 10^{+104}:\\ \;\;\;\;-\frac{m \cdot m}{m}\\ \mathbf{elif}\;m \cdot \left(-1 + \frac{m \cdot \left(1 - m\right)}{v}\right) \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))