a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.6%

Time: 10.6s

Alternatives: 10

Speedup: 1.0×

• 40.9% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65e-28) (* m (+ -1.0 (/ m v))) (/ (* m (- 1.0 m)) (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-28) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d-28) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (m * (1.0d0 - m)) / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-28) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (m * (1.0 - m)) / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-28:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (m * (1.0 - m)) / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-28)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65e-28)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (m * (1.0 - m)) / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-28], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999994e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 99.9%

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

   if 2.64999999999999994e-28 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*l/ 99.8%

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

      3. clear-num 99.8%

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

      4. associate-*l* 99.8%

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

      5. div-inv 99.9%

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

      6. associate-*l/ 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.65e-28) (* m (+ -1.0 (/ m v))) (* (- 1.0 m) (/ m (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-28) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (m / (v / m));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.65d-28) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 - m) * (m / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.65e-28) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) * (m / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.65e-28:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (1.0 - m) * (m / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.65e-28)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.65e-28)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (1.0 - m) * (m / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.65e-28], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.64999999999999994e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 99.9%

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

   if 2.64999999999999994e-28 < m

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-/r/ 99.9%

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

      4. add-cube-cbrt 99.4%

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

      5. cbrt-prod 99.7%

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

      6. unpow2 99.7%

         \[\leadsto \left(1 - m\right) \cdot \frac{\sqrt[3]{\color{blue}{{m}^{2}}} \cdot \sqrt[3]{m}}{\frac{v}{m}} \]

      7. associate-/l* 99.7%

         \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(\sqrt[3]{{m}^{2}} \cdot \frac{\sqrt[3]{m}}{\frac{v}{m}}\right)} \]

      8. unpow2 99.7%

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

      9. cbrt-prod 99.5%

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

      10. pow2 99.5%

          \[\leadsto \left(1 - m\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{m}\right)}^{2}} \cdot \frac{\sqrt[3]{m}}{\frac{v}{m}}\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left({\left(\sqrt[3]{m}\right)}^{2} \cdot \frac{\sqrt[3]{m}}{\frac{v}{m}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.4%

          \[\leadsto \left(1 - m\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{m}\right)}^{2} \cdot \sqrt[3]{m}}{\frac{v}{m}}} \]

       2. unpow2 99.4%

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

       3. rem-3cbrt-lft 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.65 \cdot 10^{-28}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.1 \cdot 10^{-18}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.1e-18) (* m (+ -1.0 (/ m v))) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.1e-18) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 6.1d-18) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 6.1e-18) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.1e-18:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.1e-18)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 6.1e-18)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.1e-18], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 6.0999999999999999e-18

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

   if 6.0999999999999999e-18 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 99.9%

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

   4. Taylor expanded in m around inf 99.8%

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

      1. cube-mult 99.9%

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

      2. unpow2 99.9%

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

      3. associate-/r* 99.9%

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

      4. div-sub 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-*r* 99.8%

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

      8. unpow2 99.8%

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

      9. associate-*r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. distribute-rgt-in 99.8%

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

      12. lft-mult-inverse 99.8%

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

      13. neg-mul-1 99.8%

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

      14. sub-neg 99.8%

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

      15. div-sub 99.8%

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

      16. unsub-neg 99.8%

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

      17. mul-1-neg 99.8%

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

      18. +-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.1 \cdot 10^{-18}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-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 99.0%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.8%

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

      4. fmm-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   8. Simplified 5.7%

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

4. Final simplification 54.1%

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

Alternative 5: 52.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(N[(m - v), $MachinePrecision] / v), $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. 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* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 98.9%

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

   6. Taylor expanded in v around 0 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

   8. Simplified 99.0%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.8%

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

      4. fmm-def 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   8. Simplified 5.7%

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

Alternative 6: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r/ 99.8%

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

   4. clear-num 99.7%

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

   5. un-div-inv 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto m \cdot \left(\color{blue}{\frac{1 - m}{\frac{v}{m}}} + -1\right) \]
7. 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 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 27.6% 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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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. *-commutative 99.8%

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

   3. associate-/l* 99.8%

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

   4. fmm-def 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around 0 99.8%

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

6. Taylor expanded in m around 0 30.7%

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

   1. neg-mul-1 30.7%

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

8. Simplified 30.7%

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

Alternative 10: 3.1% accurate, 11.0× 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 m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/l* 99.8%

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

   4. fmm-def 99.8%

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

   5. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 30.7%

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

   1. *-commutative 30.7%

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

   2. neg-mul-1 30.7%

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

   3. neg-sub0 30.7%

      \[\leadsto \color{blue}{0 - m} \]

   4. sub-neg 30.7%

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

   5. add-sqr-sqrt 0.0%

      \[\leadsto 0 + \color{blue}{\sqrt{-m} \cdot \sqrt{-m}} \]

   6. sqrt-unprod 3.4%

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

   7. sqr-neg 3.4%

      \[\leadsto 0 + \sqrt{\color{blue}{m \cdot m}} \]

   8. sqrt-unprod 3.2%

      \[\leadsto 0 + \color{blue}{\sqrt{m} \cdot \sqrt{m}} \]

   9. add-sqr-sqrt 3.2%

      \[\leadsto 0 + \color{blue}{m} \]

7. Applied egg-rr 3.2%

   \[\leadsto \color{blue}{0 + m} \]

8. Taylor expanded in m around 0 3.2%

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

Reproduce

herbie shell --seed 2024170 
(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))