a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 98.0%

Time: 7.9s

Alternatives: 11

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: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-81) (- (/ m (/ v m)) m) (* (* m (- 1.0 m)) (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-81) {
		tmp = (m / (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 <= 3.7d-81) then
        tmp = (m / (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 <= 3.7e-81) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (1.0 - m)) * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.7e-81:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * (1.0 - m)) * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-81)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.7e-81)
		tmp = (m / (v / m)) - m;
	else
		tmp = (m * (1.0 - m)) * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-81], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.69999999999999986e-81

   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. distribute-rgt-in 99.8%

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

      2. associate-*r/ 99.8%

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

      3. div-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r/ 99.8%

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

      6. clear-num 99.8%

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

      7. neg-mul-1 99.8%

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

      8. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-*r/ 86.0%

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

      5. unpow2 86.0%

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

      6. *-commutative 86.0%

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

      7. mul-1-neg 86.0%

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

      8. unsub-neg 86.0%

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

   9. Simplified 86.0%

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

       1. unpow2 86.0%

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

       2. associate-/l* 99.8%

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

       3. clear-num 99.8%

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

       4. div-inv 99.9%

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

       5. associate-/r/ 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. un-div-inv 99.9%

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

   13. Applied egg-rr 99.9%

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

   if 3.69999999999999986e-81 < 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. unpow2 99.8%

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

      3. associate-*r* 99.8%

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

      4. associate-*l/ 99.8%

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

      5. associate-*r/ 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 99.9%

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

      8. *-commutative 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[m_, v_] := N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(m / v), $MachinePrecision] + (-m)), $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. distribute-rgt-in 99.8%

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

   2. associate-*r/ 99.8%

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

   3. div-inv 99.8%

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

   4. associate-*l* 99.8%

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

   5. associate-/r/ 99.9%

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

   6. clear-num 99.9%

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

   7. neg-mul-1 99.9%

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

   8. fma-define 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 3: 38.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-148} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 1.85e-148) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 1.85e-148) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if ((m <= 1.85d-148) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m * (m / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m <= 1.85e-148) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 1.85e-148) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 1.85e-148) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m * Float64(m / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 1.85e-148) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 1.85e-148], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.85000000000000017e-148 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 31.1%

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

      1. neg-mul-1 31.1%

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

   7. Simplified 31.1%

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

   if 1.85000000000000017e-148 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 80.2%

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

      1. associate-/l* 80.2%

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

      2. unpow2 80.2%

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

      3. associate-*l* 80.2%

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

      4. *-commutative 80.2%

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

      5. associate-*r* 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in m around 0 74.6%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-148} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-81) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-81) {
		tmp = (m / (v / m)) - m;
	} 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 <= 3.7d-81) then
        tmp = (m / (v / m)) - m
    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 <= 3.7e-81) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.7e-81:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-81)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	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 <= 3.7e-81)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-81], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $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 < 3.69999999999999986e-81

   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. distribute-rgt-in 99.8%

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

      2. associate-*r/ 99.8%

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

      3. div-inv 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-/r/ 99.8%

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

      6. clear-num 99.8%

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

      7. neg-mul-1 99.8%

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

      8. fma-define 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 99.8%

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

      1. sub-neg 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. associate-*r/ 86.0%

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

      5. unpow2 86.0%

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

      6. *-commutative 86.0%

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

      7. mul-1-neg 86.0%

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

      8. unsub-neg 86.0%

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

   9. Simplified 86.0%

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

       1. unpow2 86.0%

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

       2. associate-/l* 99.8%

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

       3. clear-num 99.8%

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

       4. div-inv 99.9%

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

       5. associate-/r/ 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. clear-num 99.8%

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

       3. un-div-inv 99.9%

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

   13. Applied egg-rr 99.9%

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

   if 3.69999999999999986e-81 < 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. associate-/l* 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*r* 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      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. Step-by-step derivation

      1. distribute-rgt-in 99.7%

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

      2. associate-*r/ 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r/ 99.8%

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

      6. clear-num 99.8%

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

      7. neg-mul-1 99.8%

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

      8. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0 97.1%

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

      1. sub-neg 97.1%

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

      2. metadata-eval 97.1%

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

      3. distribute-lft-in 97.1%

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

      4. associate-*r/ 86.5%

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

      5. unpow2 86.5%

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

      6. *-commutative 86.5%

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

      7. mul-1-neg 86.5%

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

      8. unsub-neg 86.5%

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

   9. Simplified 86.5%

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

       1. unpow2 86.5%

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

       2. associate-/l* 97.1%

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

       3. clear-num 97.0%

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

       4. div-inv 97.1%

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

       5. associate-/r/ 97.1%

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

   11. Applied egg-rr 97.1%

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

       1. *-commutative 97.1%

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

       2. clear-num 97.0%

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

       3. un-div-inv 97.1%

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

   13. Applied egg-rr 97.1%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in m around inf 98.5%

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

      1. neg-mul-1 98.5%

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

      2. distribute-neg-frac2 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 97.8%

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

Alternative 6: 51.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $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. Step-by-step derivation

      1. distribute-rgt-in 99.7%

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

      2. associate-*r/ 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.7%

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

      5. associate-/r/ 99.8%

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

      6. clear-num 99.8%

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

      7. neg-mul-1 99.8%

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

      8. fma-define 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0 97.1%

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

      1. sub-neg 97.1%

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

      2. metadata-eval 97.1%

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

      3. distribute-lft-in 97.1%

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

      4. associate-*r/ 86.5%

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

      5. unpow2 86.5%

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

      6. *-commutative 86.5%

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

      7. mul-1-neg 86.5%

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

      8. unsub-neg 86.5%

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

   9. Simplified 86.5%

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

       1. unpow2 86.5%

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

       2. associate-/l* 97.1%

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

       3. clear-num 97.0%

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

       4. div-inv 97.1%

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

       5. associate-/r/ 97.1%

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

   11. Applied egg-rr 97.1%

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

       1. *-commutative 97.1%

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

       2. clear-num 97.0%

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

       3. un-div-inv 97.1%

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

   13. Applied egg-rr 97.1%

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

   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%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 5.6%

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

      1. neg-mul-1 5.6%

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

   7. Simplified 5.6%

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

Alternative 7: 51.3% 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 97.1%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 5.6%

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

      1. neg-mul-1 5.6%

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

   7. Simplified 5.6%

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

4. Final simplification 51.3%

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

Alternative 8: 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 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 9: 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/ 99.9%

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

   4. clear-num 99.9%

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

   5. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 10: 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 11: 27.1% 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%

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

   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 m around 0 28.1%

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

   1. neg-mul-1 28.1%

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

7. Simplified 28.1%

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

Reproduce

herbie shell --seed 2024090 
(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))