a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 9

Speedup: 0.1×

• 42.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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. distribute-rgt-in 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*r/ 99.8%

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

   4. *-commutative 99.8%

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

   5. associate-*r/ 99.9%

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

   6. neg-mul-1 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(\left(1 - m\right) \cdot \frac{m}{v}, m, -m\right) \]
8. Add Preprocessing

Alternative 2: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.6e-148) (- m) (if (<= m 1.0) (* m (/ m v)) (* (/ m v) (- m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.6e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (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 <= 4.6d-148) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = (m / v) * -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.6e-148) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.6e-148:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.6e-148)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.6e-148)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.6e-148], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 4.59999999999999995e-148

   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 74.8%

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

      1. neg-mul-1 74.8%

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

   7. Simplified 74.8%

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

   if 4.59999999999999995e-148 < m < 1

   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 0 96.9%

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

   4. Taylor expanded in m around inf 80.8%

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

   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 0.1%

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

   6. Taylor expanded in m around inf 0.1%

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

      1. unpow2 0.1%

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

      2. associate-/l* 0.1%

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

      3. clear-num 0.1%

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

      4. metadata-eval 0.1%

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

      5. associate-*l/ 0.1%

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

      6. associate-*l* 0.1%

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

      7. metadata-eval 0.1%

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

      8. distribute-neg-frac 0.1%

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

      9. clear-num 0.1%

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

      10. distribute-frac-neg2 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. *-commutative 0.1%

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

      14. distribute-frac-neg2 0.1%

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

      15. clear-num 0.1%

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

      16. distribute-neg-frac 0.1%

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

      17. metadata-eval 0.1%

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

      18. associate-*l* 0.1%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-148}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.8 \cdot 10^{-146} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 9.8e-146) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 9.8e-146) || !(m <= 1.0)) {
		tmp = -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) :: tmp
    if ((m <= 9.8d-146) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m <= 9.8e-146) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 9.8e-146) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 9.8e-146) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 9.8e-146) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 9.8e-146], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 9.8000000000000008e-146 or 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 26.0%

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

      1. neg-mul-1 26.0%

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

   7. Simplified 26.0%

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

   if 9.8000000000000008e-146 < m < 1

   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 0 96.9%

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

   4. Taylor expanded in m around inf 80.8%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.8 \cdot 10^{-146} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-10) (* m (+ (/ m v) -1.0)) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-10) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m * (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 <= 2.1d-10) then
        tmp = m * ((m / v) + (-1.0d0))
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-10) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-10:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-10)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-10)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-10], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.1e-10

   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 99.5%

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

   if 2.1e-10 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-10) (* m (+ (/ m v) -1.0)) (* m (* (- 1.0 m) (/ m v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-10) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 2.1d-10) then
        tmp = m * ((m / v) + (-1.0d0))
    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 <= 2.1e-10) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m * ((1.0 - m) * (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-10:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m * ((1.0 - m) * (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-10)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) * Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-10)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = m * ((1.0 - m) * (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-10], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.1e-10

   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 99.5%

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

   if 2.1e-10 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-*l/ 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-10}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $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 98.4%

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

   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 0.1%

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

   6. Taylor expanded in m around inf 0.1%

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

      1. unpow2 0.1%

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

      2. associate-/l* 0.1%

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

      3. clear-num 0.1%

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

      4. metadata-eval 0.1%

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

      5. associate-*l/ 0.1%

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

      6. associate-*l* 0.1%

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

      7. metadata-eval 0.1%

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

      8. distribute-neg-frac 0.1%

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

      9. clear-num 0.1%

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

      10. distribute-frac-neg2 0.1%

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

      11. add-sqr-sqrt 0.0%

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

      12. add-sqr-sqrt 0.1%

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

      13. *-commutative 0.1%

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

      14. distribute-frac-neg2 0.1%

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

      15. clear-num 0.1%

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

      16. distribute-neg-frac 0.1%

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

      17. metadata-eval 0.1%

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

      18. associate-*l* 0.1%

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

   8. Applied egg-rr 75.5%

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

4. Final simplification 86.0%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

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(m * Float64(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[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 26.2% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   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 23.2%

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

   1. neg-mul-1 23.2%

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

7. Simplified 23.2%

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

8. Final simplification 23.2%

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

Reproduce

herbie shell --seed 2024074 
(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))