a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

Alternatives: 7

Speedup: N/A×

• 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.8%

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

   1. neg-mul-1 99.8%

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

   2. +-commutative 99.8%

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

   3. sub-neg 99.8%

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

   4. div-sub 99.8%

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

   5. associate-*r/ 99.9%

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

   6. associate-*l/ 99.9%

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

   7. associate-/r/ 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.0%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around inf 98.1%

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

      1. neg-mul-1 98.1%

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

      2. distribute-neg-frac2 98.1%

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

   7. Simplified 98.1%

      \[\leadsto m \cdot \left(m \cdot \color{blue}{\frac{m}{-v}} + -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (-1.0 - (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 <= 1.0d0) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * ((-1.0d0) - (m / (v / m)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.0%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. mul-1-neg 98.2%

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

   10. Simplified 98.2%

       \[\leadsto m \cdot \left(\frac{m}{\color{blue}{\frac{-v}{m}}} + -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

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

Alternative 4: 87.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.0%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.7%

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

      2. pow3 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow3 0.1%

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

      2. add-cube-cbrt 0.1%

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

      3. div-inv 0.1%

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

      4. add-sqr-sqrt 0.1%

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

      5. sqrt-prod 0.1%

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

      6. unpow2 0.1%

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

      7. add-sqr-sqrt 0.1%

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

      8. sqrt-unprod 0.1%

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

      9. unpow2 0.1%

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

      10. sqr-neg 0.1%

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

      11. unpow2 0.1%

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

      12. swap-sqr 0.1%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 0.0%

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

   9. Applied egg-rr 77.7%

      \[\leadsto m \cdot \left(\color{blue}{\left(0 - \frac{m}{v}\right)} + -1\right) \]
   10. Step-by-step derivation

       1. neg-sub0 77.7%

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

       2. distribute-frac-neg2 77.7%

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

   11. Simplified 77.7%

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

   12. Taylor expanded in m around 0 77.7%

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

       1. sub-neg 77.7%

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

       2. metadata-eval 77.7%

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

       3. +-commutative 77.7%

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

       4. neg-mul-1 77.7%

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

       5. unsub-neg 77.7%

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

   14. Simplified 77.7%

       \[\leadsto \color{blue}{m \cdot \left(-1 - \frac{m}{v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 62.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. add-cube-cbrt 99.4%

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

   2. pow3 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in m around 0 47.6%

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

   1. unpow3 47.6%

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

   2. add-cube-cbrt 47.9%

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

   3. div-inv 47.9%

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

   4. add-sqr-sqrt 47.8%

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

   5. sqrt-prod 42.0%

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

   6. unpow2 42.0%

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

   7. add-sqr-sqrt 42.0%

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

   8. sqrt-unprod 31.4%

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

   9. unpow2 31.4%

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

   10. sqr-neg 31.4%

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

   11. unpow2 31.4%

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

   12. swap-sqr 31.4%

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

   13. sqrt-unprod 15.1%

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

   14. add-sqr-sqrt 15.1%

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

9. Applied egg-rr 63.2%

   \[\leadsto m \cdot \left(\color{blue}{\left(0 - \frac{m}{v}\right)} + -1\right) \]
10. Step-by-step derivation

    1. neg-sub0 63.2%

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

    2. distribute-frac-neg2 63.2%

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

11. Simplified 63.2%

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

12. Taylor expanded in m around 0 63.2%

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

    1. sub-neg 63.2%

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

    2. metadata-eval 63.2%

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

    3. +-commutative 63.2%

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

    4. neg-mul-1 63.2%

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

    5. unsub-neg 63.2%

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

14. Simplified 63.2%

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

15. Final simplification 63.2%

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

Alternative 7: 26.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ -m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- m))

C

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = -m
end function

Java

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

Python

def code(m, v):
	return -m

Julia

function code(m, v)
	return Float64(-m)
end

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 26.4%

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

   1. neg-mul-1 26.4%

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

7. Simplified 26.4%

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

8. Final simplification 26.4%

   \[\leadsto -m \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))