a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

Alternatives: 11

Speedup: 1.0×

• 41.8% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. associate-*r/ 99.8%

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

   2. clear-num 99.8%

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

6. Applied egg-rr 99.8%

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

   1. distribute-lft-in 99.8%

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

   2. un-div-inv 99.8%

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

   3. associate-/r* 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. neg-mul-1 99.9%

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

   3. unsub-neg 99.9%

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

   4. associate-/r/ 99.9%

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

10. Simplified 99.9%

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

Alternative 2: 38.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-166} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 2.5e-166) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 2.5e-166) || !(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 <= 2.5d-166) .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 <= 2.5e-166) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 2.5e-166) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 2.5e-166) || !(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 <= 2.5e-166) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 2.5e-166], 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 < 2.5e-166 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 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around 0 32.9%

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

      1. neg-mul-1 32.9%

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

   10. Simplified 32.9%

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

   if 2.5e-166 < m < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in m around 0 95.4%

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

   4. Taylor expanded in m around inf 73.6%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-166} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m}{\frac{v}{m}}\\ \mathbf{if}\;m \leq 1:\\ \;\;\;\;t\_0 - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (/ m (/ v m)))) (if (<= m 1.0) (- t_0 m) (* m (- -1.0 t_0)))))

C

double code(double m, double v) {
	double t_0 = m / (v / m);
	double tmp;
	if (m <= 1.0) {
		tmp = t_0 - m;
	} else {
		tmp = m * (-1.0 - t_0);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m / (v / m)
    if (m <= 1.0d0) then
        tmp = t_0 - m
    else
        tmp = m * ((-1.0d0) - t_0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double t_0 = m / (v / m);
	double tmp;
	if (m <= 1.0) {
		tmp = t_0 - m;
	} else {
		tmp = m * (-1.0 - t_0);
	}
	return tmp;
}

Python

def code(m, v):
	t_0 = m / (v / m)
	tmp = 0
	if m <= 1.0:
		tmp = t_0 - m
	else:
		tmp = m * (-1.0 - t_0)
	return tmp

Julia

function code(m, v)
	t_0 = Float64(m / Float64(v / m))
	tmp = 0.0
	if (m <= 1.0)
		tmp = Float64(t_0 - m);
	else
		tmp = Float64(m * Float64(-1.0 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	t_0 = m / (v / m);
	tmp = 0.0;
	if (m <= 1.0)
		tmp = t_0 - m;
	else
		tmp = m * (-1.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := Block[{t$95$0 = N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 1.0], N[(t$95$0 - m), $MachinePrecision], N[(m * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if 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. Step-by-step derivation

      1. associate-*r/ 99.7%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. un-div-inv 99.8%

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

      3. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in m around 0 97.7%

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

   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 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. mul-1-neg 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 98.1%

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

Alternative 4: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		tmp = (m / (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 / 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 / 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 / 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 / m), $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 99.7%

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

      1. associate-*r/ 99.7%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. un-div-inv 99.8%

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

      3. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in m around 0 97.7%

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

   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 inf 98.5%

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

      1. neg-mul-1 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.1%

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

Alternative 5: 87.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (m / (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 / 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 / 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 / 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 / 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 / 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 / m), $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 99.7%

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

      1. associate-*r/ 99.7%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. un-div-inv 99.8%

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

      3. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in m around 0 97.7%

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

   if 1 < 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 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. neg-sub0 0.0%

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

      4. sqrt-unprod 79.8%

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

      5. sqr-neg 79.8%

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

      6. sqrt-unprod 79.2%

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

      7. add-sqr-sqrt 79.2%

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

      8. div-sub 79.2%

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

   5. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\left(\frac{0}{v} - \frac{m}{v}\right)} - 1\right) \cdot m \]
   6. Step-by-step derivation

      1. div0 79.2%

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

      2. neg-sub0 79.2%

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

      3. distribute-neg-frac2 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 88.9%

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

Alternative 6: 87.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * (-1.0 + (m / v))
	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(-1.0 + Float64(m / v)));
	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 * (-1.0 + (m / v));
	else
		tmp = m * (-1.0 - (m / v));
	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], 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 99.7%

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

   3. Taylor expanded in m around 0 97.7%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. neg-sub0 0.0%

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

      4. sqrt-unprod 79.8%

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

      5. sqr-neg 79.8%

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

      6. sqrt-unprod 79.2%

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

      7. add-sqr-sqrt 79.2%

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

      8. div-sub 79.2%

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

   5. Applied egg-rr 79.2%

      \[\leadsto \left(\color{blue}{\left(\frac{0}{v} - \frac{m}{v}\right)} - 1\right) \cdot m \]
   6. Step-by-step derivation

      1. div0 79.2%

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

      2. neg-sub0 79.2%

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

      3. distribute-neg-frac2 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 88.8%

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

Alternative 7: 51.3% accurate, 0.9× speedup

Math

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

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 99.7%

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

   3. Taylor expanded in m around 0 97.7%

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

   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 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. associate-*r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-/r/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in m around 0 5.7%

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

      1. neg-mul-1 5.7%

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

   10. Simplified 5.7%

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

4. Final simplification 53.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

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(m / Float64(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[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $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* 99.7%

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

   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 99.7%

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

   1. neg-mul-1 99.7%

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

   2. +-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.7%

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

   5. associate-*r/ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

Alternative 9: 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* 99.7%

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

   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. Final simplification 99.7%

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

Alternative 10: 27.0% 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* 99.7%

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

   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 99.7%

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

   1. neg-mul-1 99.7%

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

   2. +-commutative 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.7%

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

   5. associate-*r/ 99.8%

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

   6. associate-*l/ 99.8%

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

   7. associate-/r/ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in m around 0 29.2%

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

   1. neg-mul-1 29.2%

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

10. Simplified 29.2%

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

Alternative 11: 3.0% 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. 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 29.2%

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

   1. *-commutative 29.2%

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

   2. neg-mul-1 29.2%

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

   3. neg-sub0 29.2%

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

   4. sub-neg 29.2%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 3.5%

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

   7. sqr-neg 3.5%

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

   8. sqrt-unprod 3.4%

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

   9. add-sqr-sqrt 3.4%

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

7. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{0 + m} \]
8. Step-by-step derivation

   1. +-lft-identity 3.4%

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

9. Simplified 3.4%

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

Reproduce

herbie shell --seed 2024150 
(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))