a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 2: 88.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   5. Taylor expanded in v around 0 86.0%

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

      1. neg-mul-1 86.0%

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

      2. associate-*r/ 85.2%

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

      3. +-commutative 85.2%

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

      4. unsub-neg 85.2%

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

   7. Simplified 85.2%

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

      1. associate-*r/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-*l* 85.9%

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

      4. associate-*r/ 99.8%

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

      5. clear-num 99.6%

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

      6. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in m around 0 99.0%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. associate-*l/ 0.1%

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

      2. add-sqr-sqrt 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sqrt-unprod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 68.6%

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

      8. clear-num 68.6%

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

      9. frac-2neg 68.6%

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

      10. metadata-eval 68.6%

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

      11. distribute-frac-neg 68.6%

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

      12. remove-double-neg 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-/l* 68.6%

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

       2. neg-mul-1 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 83.9%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   5. Taylor expanded in v around 0 86.0%

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

      1. neg-mul-1 86.0%

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

      2. associate-*r/ 85.2%

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

      3. +-commutative 85.2%

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

      4. unsub-neg 85.2%

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

   7. Simplified 85.2%

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

      1. associate-*r/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-*l* 85.9%

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

      4. associate-*r/ 99.8%

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

      5. clear-num 99.6%

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

      6. un-div-inv 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in m around 0 99.0%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf 96.4%

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

      1. neg-mul-1 96.4%

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

      2. distribute-neg-frac 96.4%

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

   9. Simplified 96.4%

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

4. Final simplification 97.7%

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

Alternative 4: 51.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 98.0%

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

   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. Add Preprocessing

   5. Taylor expanded in m around 0 5.1%

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

      1. neg-mul-1 5.1%

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

   7. Simplified 5.1%

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

4. Final simplification 51.9%

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

Alternative 5: 87.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 98.0%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

      3. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. associate-*l/ 0.1%

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

      2. add-sqr-sqrt 0.1%

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

      3. *-un-lft-identity 0.1%

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

      4. sqrt-unprod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 68.6%

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

      8. clear-num 68.6%

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

      9. frac-2neg 68.6%

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

      10. metadata-eval 68.6%

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

      11. distribute-frac-neg 68.6%

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

      12. remove-double-neg 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-/l* 68.6%

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

       2. neg-mul-1 68.6%

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

   11. Simplified 68.6%

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

4. Final simplification 83.4%

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

Alternative 6: 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. 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. Add Preprocessing

5. Taylor expanded in m around 0 24.1%

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

   1. neg-mul-1 24.1%

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

7. Simplified 24.1%

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

8. Final simplification 24.1%

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

Reproduce

herbie shell --seed 2024040 
(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))