a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 11

Speedup: 1.1×

• 41.2% 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 67.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 5.7

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

   5. Applied rewrites 5.7%

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

   7. Applied rewrites 60.2%

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

   if -inf.0 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 #s(literal 1 binary64) m)) v) #s(literal 1 binary64)) m) < -1.9999999999999998e-308

   1. Initial program 99.8%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   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

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

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

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

      2. *-commutative N/A

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

      3. cube-mult N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. rgt-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. cube-mult N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in m around 0

      \[\leadsto \frac{{m}^{2}}{v} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.3%

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

4. Final simplification 67.3%

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

Reproduce

herbie shell --seed 2024219 
(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))