b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 15

Speedup: N/A×

• 41.5% 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

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (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) * (1.0d0 - m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{m + m \cdot \left(m \cdot \left(m + -2\right)\right)}{v} + -1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (+ (/ (+ m (* m (* m (+ m -2.0)))) v) -1.0))

C

double code(double m, double v) {
	return ((m + (m * (m * (m + -2.0)))) / v) + -1.0;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = ((m + (m * (m * (m + (-2.0d0))))) / v) + (-1.0d0)
end function

Java

public static double code(double m, double v) {
	return ((m + (m * (m * (m + -2.0)))) / v) + -1.0;
}

Python

def code(m, v):
	return ((m + (m * (m * (m + -2.0)))) / v) + -1.0

Julia

function code(m, v)
	return Float64(Float64(Float64(m + Float64(m * Float64(m * Float64(m + -2.0)))) / v) + -1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = ((m + (m * (m * (m + -2.0)))) / v) + -1.0;
end

Wolfram

code[m_, v_] := N[(N[(N[(m + N[(m * N[(m * N[(m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 99.7%

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

6. Taylor expanded in v around 0 99.9%

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

   1. distribute-rgt-in 99.9%

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

   2. *-un-lft-identity 99.9%

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

   3. sub-neg 99.9%

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

   4. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0);
	else
		tmp = Float64(m * Float64(Float64(v + Float64(m * Float64(m + -1.0))) / 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) + -1.0;
	else
		tmp = m * ((v + (m * (m + -1.0))) / v);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 95.8%

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

   if 1 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

      4. distribute-lft-in 50.4%

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

      5. associate-*r/ 50.4%

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

      6. associate-*l/ 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-in 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in v around 0 96.6%

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

4. Final simplification 96.2%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 95.8%

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

   if 1 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

      4. distribute-lft-in 50.4%

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

      5. associate-*r/ 50.4%

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

      6. associate-*l/ 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-in 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in v around 0 96.6%

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

4. Final simplification 96.2%

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

Alternative 4: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 95.8%

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

   if 1 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

      4. distribute-lft-in 50.4%

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

      5. associate-*r/ 50.4%

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

      6. associate-*l/ 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-in 96.5%

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

   8. Simplified 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 96.5%

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

      3. un-div-inv 96.6%

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

   10. Applied egg-rr 96.6%

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

4. Final simplification 96.2%

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

Alternative 5: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 95.8%

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

   if 1 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf 96.6%

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

      1. neg-mul-1 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-neg-frac 96.6%

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

      3. metadata-eval 96.6%

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

      4. distribute-lft-in 50.4%

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

      5. associate-*r/ 50.4%

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

      6. associate-*l/ 50.4%

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

      7. *-commutative 50.4%

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

      8. distribute-rgt-in 96.5%

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

   8. Simplified 96.5%

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

4. Final simplification 96.2%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{m \cdot \left(1 + m \cdot \left(m - 2\right)\right)}{v} + -1 \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (+ (/ (* m (+ 1.0 (* m (- m 2.0)))) v) -1.0))

C

double code(double m, double v) {
	return ((m * (1.0 + (m * (m - 2.0)))) / 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 - 2.0d0)))) / v) + (-1.0d0)
end function

Java

public static double code(double m, double v) {
	return ((m * (1.0 + (m * (m - 2.0)))) / v) + -1.0;
}

Python

def code(m, v):
	return ((m * (1.0 + (m * (m - 2.0)))) / v) + -1.0

Julia

function code(m, v)
	return Float64(Float64(Float64(m * Float64(1.0 + Float64(m * Float64(m - 2.0)))) / v) + -1.0)
end

MATLAB

function tmp = code(m, v)
	tmp = ((m * (1.0 + (m * (m - 2.0)))) / v) + -1.0;
end

Wolfram

code[m_, v_] := N[(N[(N[(m * N[(1.0 + N[(m * N[(m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 99.7%

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

6. Taylor expanded in v around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (1.0 - 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 = (1.0d0 - m) * ((m / (v / (1.0d0 - m))) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. metadata-eval 99.7%

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

   2. sub-neg 99.7%

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

   3. clear-num 99.7%

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

   4. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (1.0 - 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 = (1.0d0 - m) * ((m * ((1.0d0 - m) / v)) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

Alternative 9: 97.9% accurate, 1.2× speedup

Math

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

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 99.7%

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

6. Taylor expanded in v around 0 99.9%

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

7. Taylor expanded in m around inf 96.1%

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

8. Final simplification 96.1%

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

Alternative 10: 86.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return ((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 / v) + (-1.0d0)) * (m + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 47.1%

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

   1. *-commutative 47.1%

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

   2. sub-neg 47.1%

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

   3. distribute-rgt-in 47.1%

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

   4. *-un-lft-identity 47.1%

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

   5. un-div-inv 47.2%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 84.0%

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

   8. sqr-neg 84.0%

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

   9. sqrt-unprod 84.0%

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

   10. add-sqr-sqrt 84.0%

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

   11. un-div-inv 84.0%

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

7. Applied egg-rr 84.0%

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

   1. distribute-rgt1-in 84.0%

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

9. Simplified 84.0%

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

10. Final simplification 84.0%

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

Alternative 11: 75.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	return (m + (m / v)) + -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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 99.7%

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

6. Taylor expanded in m around 0 72.4%

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

   1. distribute-rgt-in 72.4%

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

   2. *-lft-identity 72.4%

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

   3. associate-*l/ 72.6%

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

   4. *-lft-identity 72.6%

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

8. Simplified 72.6%

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

9. Final simplification 72.6%

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

Alternative 12: 26.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.32 \cdot 10^{-52}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 1.32e-52) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.32e-52) {
		tmp = -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 <= 1.32d-52) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.32e-52) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.32e-52:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.32e-52)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.32e-52)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.32e-52], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 1.32000000000000002e-52

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 53.5%

      \[\leadsto \color{blue}{-1} \]

   if 1.32000000000000002e-52 < m

   1. Initial program 99.8%

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

   3. Taylor expanded in m around inf 81.1%

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

      1. neg-mul-1 81.1%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in m around 0 5.2%

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

Alternative 13: 75.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{m}{v} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 99.7%

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

6. Taylor expanded in v around 0 99.9%

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

7. Taylor expanded in m around 0 72.6%

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

8. Final simplification 72.6%

   \[\leadsto \frac{m}{v} + -1 \]
9. Add Preprocessing

Alternative 14: 27.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around inf 23.9%

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

   1. neg-mul-1 23.9%

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

   2. neg-sub0 23.9%

      \[\leadsto \color{blue}{0 - \left(1 - m\right)} \]

   3. associate--r- 23.9%

      \[\leadsto \color{blue}{\left(0 - 1\right) + m} \]

   4. metadata-eval 23.9%

      \[\leadsto \color{blue}{-1} + m \]

7. Simplified 23.9%

   \[\leadsto \color{blue}{-1 + m} \]

8. Final simplification 23.9%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 15: 24.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in m around 0 21.6%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024156 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))