a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 9

Speedup: 1.0×

• 42.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(m * N[(N[(1.0 - m), $MachinePrecision] * N[(m / 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. 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. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. fma-neg 99.9%

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

   5. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 37.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-170} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 9.6e-170) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 9.6e-170) || !(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 <= 9.6d-170) .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 <= 9.6e-170) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 9.6e-170) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 9.6e-170) || !(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 <= 9.6e-170) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 9.6e-170], 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 < 9.5999999999999998e-170 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. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. flip-- 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow2 99.9%

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

      8. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. unpow2 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. *-lft-identity 99.9%

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

      7. distribute-rgt-neg-out 99.9%

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

      8. unpow2 99.9%

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

      9. distribute-lft-neg-in 99.9%

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

      10. unpow2 99.9%

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

      11. unpow3 99.9%

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

      12. cube-neg 99.9%

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

      13. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around 0 99.9%

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

       1. mul-1-neg 99.9%

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

       2. distribute-rgt-in 99.9%

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

       3. *-lft-identity 99.9%

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

       4. cancel-sign-sub-inv 99.9%

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

       5. unpow2 99.9%

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

       6. unpow3 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in m around 0 29.3%

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

       1. mul-1-neg 29.3%

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

   14. Simplified 29.3%

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

   if 9.5999999999999998e-170 < m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0 97.8%

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

   4. Taylor expanded in m around inf 73.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 9.6 \cdot 10^{-170} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	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 * ((m / v) + -1.0);
	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[(N[(m / v), $MachinePrecision] + -1.0), $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 98.7%

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

   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 96.7%

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

      1. neg-mul-1 96.7%

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

   7. Simplified 96.7%

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

4. Final simplification 97.7%

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

Alternative 4: 87.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m * (-1.0 + (m * (-1.0 / v)))
	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(m * Float64(-1.0 + Float64(m * Float64(-1.0 / v))));
	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 * (-1.0 + (m * (-1.0 / v)));
	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[(m * N[(-1.0 + N[(m * N[(-1.0 / 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 98.7%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. metadata-eval 0.1%

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

      3. div-inv 0.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 78.7%

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

      6. sqr-neg 78.7%

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

      7. sqrt-unprod 76.1%

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

      8. add-sqr-sqrt 76.1%

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

   7. Applied egg-rr 76.1%

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

      1. associate-*r/ 76.1%

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

      2. metadata-eval 76.1%

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

   9. Simplified 76.1%

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

4. Final simplification 87.0%

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

Alternative 5: 50.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

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 98.7%

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

   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. associate-*r/ 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. flip-- 99.9%

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

      5. frac-times 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow2 99.9%

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

      8. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. unpow2 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. *-lft-identity 99.9%

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

      7. distribute-rgt-neg-out 99.9%

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

      8. unpow2 99.9%

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

      9. distribute-lft-neg-in 99.9%

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

      10. unpow2 99.9%

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

      11. unpow3 99.9%

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

      12. cube-neg 99.9%

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

      13. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in m around 0 99.9%

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

       1. mul-1-neg 99.9%

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

       2. distribute-rgt-in 99.9%

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

       3. *-lft-identity 99.9%

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

       4. cancel-sign-sub-inv 99.9%

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

       5. unpow2 99.9%

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

       6. unpow3 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in m around 0 5.5%

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

       1. mul-1-neg 5.5%

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

   14. Simplified 5.5%

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

4. Final simplification 50.3%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

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(Float64(m * 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[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $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%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m * N[(-1.0 + N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $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* 99.9%

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

   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. associate-*r/ 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*r/ 99.9%

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

   4. clear-num 99.9%

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

   5. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 8: 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* 99.9%

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

   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. Final simplification 99.9%

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

Alternative 9: 26.6% 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* 99.9%

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

   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. associate-*r/ 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*r/ 99.9%

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

   4. flip-- 99.9%

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

   5. frac-times 99.9%

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

   6. metadata-eval 99.9%

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

   7. pow2 99.9%

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

   8. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. unpow2 99.9%

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

   4. distribute-rgt-neg-out 99.9%

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

   5. distribute-rgt-in 99.9%

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

   6. *-lft-identity 99.9%

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

   7. distribute-rgt-neg-out 99.9%

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

   8. unpow2 99.9%

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

   9. distribute-lft-neg-in 99.9%

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

   10. unpow2 99.9%

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

   11. unpow3 99.9%

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

   12. cube-neg 99.9%

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

   13. *-commutative 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in m around 0 99.9%

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

    1. mul-1-neg 99.9%

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

    2. distribute-rgt-in 99.9%

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

    3. *-lft-identity 99.9%

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

    4. cancel-sign-sub-inv 99.9%

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

    5. unpow2 99.9%

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

    6. unpow3 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in m around 0 27.6%

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

    1. mul-1-neg 27.6%

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

14. Simplified 27.6%

    \[\leadsto \color{blue}{-m} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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))