a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.6%

Time: 5.6s

Alternatives: 6

Speedup: 1.0×

• 41.9% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2e-20) (* m (+ (/ m v) -1.0)) (/ m (/ v (* m (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-20) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (m * (1.0 - 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 <= 2d-20) then
        tmp = m * ((m / v) + (-1.0d0))
    else
        tmp = m / (v / (m * (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2e-20) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-20:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m / (v / (m * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2e-20)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m / Float64(v / Float64(m * Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2e-20)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = m / (v / (m * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2e-20], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.99999999999999989e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 99.9%

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

   if 1.99999999999999989e-20 < 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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 17.7%

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

      2. associate-*r/ 17.1%

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

      3. metadata-eval 17.1%

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

      4. sub-neg 17.1%

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

      5. pow2 17.1%

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

      6. associate-/r/ 17.0%

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

      7. metadata-eval 17.0%

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

      8. fma-neg 17.0%

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

      9. metadata-eval 17.0%

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

   5. Applied egg-rr 17.0%

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

      1. associate-/l* 17.7%

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

      2. +-commutative 17.7%

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

      3. associate-/r/ 17.7%

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

      4. *-commutative 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in v around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 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 99.9%

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

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 51.2% accurate, 1.2× 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. Taylor expanded in m around 0 98.0%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   6. Simplified 5.7%

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

4. Final simplification 48.6%

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

Alternative 5: 38.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 4.8e-148) (/ m (/ v m)) (- m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 4.8e-148) {
		tmp = m / (v / m);
	} 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 (v <= 4.8d-148) then
        tmp = m / (v / m)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 4.8e-148) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 4.8e-148:
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 4.8e-148)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 4.8e-148)
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 4.8e-148], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]

Derivation

1. Split input into 2 regimes

2. if v < 4.8000000000000002e-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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 39.1%

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

      2. associate-*r/ 39.1%

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

      3. metadata-eval 39.1%

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

      4. sub-neg 39.1%

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

      5. pow2 39.1%

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

      6. associate-/r/ 39.1%

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

      7. metadata-eval 39.1%

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

      8. fma-neg 39.1%

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

      9. metadata-eval 39.1%

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

   5. Applied egg-rr 39.1%

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

      1. associate-/l* 39.1%

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

      2. +-commutative 39.1%

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

      3. associate-/r/ 39.1%

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

      4. *-commutative 39.1%

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

   7. Simplified 39.1%

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

   8. Taylor expanded in v around 0 87.9%

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

   9. Taylor expanded in m around 0 32.4%

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

   if 4.8000000000000002e-148 < v

   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%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 40.9%

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

      1. neg-mul-1 40.9%

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

   6. Simplified 40.9%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 4.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 6: 26.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 27.0%

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

   1. neg-mul-1 27.0%

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

6. Simplified 27.0%

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

7. Final simplification 27.0%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023318 
(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))