b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 12

Speedup: 1.0×

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. metadata-eval 99.8%

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

   2. sub-neg 99.8%

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

   3. clear-num 99.8%

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

   4. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-lft-in 53.8%

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

      3. distribute-rgt-neg-in 53.8%

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

      4. associate-*r/ 53.8%

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

      5. *-rgt-identity 53.8%

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

      6. neg-mul-1 53.8%

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

      7. distribute-rgt-in 96.6%

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

   8. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 97.9%

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

Alternative 3: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-lft-in 53.8%

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

      3. distribute-rgt-neg-in 53.8%

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

      4. associate-*r/ 53.8%

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

      5. *-rgt-identity 53.8%

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

      6. neg-mul-1 53.8%

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

      7. distribute-rgt-in 96.6%

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

   8. Simplified 96.6%

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

      1. *-commutative 96.6%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 97.9%

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

Alternative 4: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.2%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in m around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. distribute-lft-in 53.8%

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

      3. distribute-rgt-neg-in 53.8%

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

      4. associate-*r/ 53.8%

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

      5. *-rgt-identity 53.8%

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

      6. neg-mul-1 53.8%

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

      7. distribute-rgt-in 96.6%

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

   8. Simplified 96.6%

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

4. Final simplification 97.9%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 6: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 + m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.4) (+ (/ m v) -1.0) (* m (/ (+ 1.0 m) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * ((1.0 + m) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.4d0) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = m * ((1.0d0 + m) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.4) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = m * ((1.0 + m) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.4:
		tmp = (m / v) + -1.0
	else:
		tmp = m * ((1.0 + m) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.4)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(m * Float64(Float64(1.0 + m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.4)
		tmp = (m / v) + -1.0;
	else
		tmp = m * ((1.0 + m) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.4], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(m * N[(N[(1.0 + m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.39999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0 99.2%

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

   4. Taylor expanded in m around 0 99.2%

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

   if 2.39999999999999991 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0 0.1%

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

      1. sub-neg 0.1%

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

      2. distribute-lft-in 0.1%

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

      3. *-commutative 0.1%

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

      4. *-un-lft-identity 0.1%

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

      5. sub-neg 0.1%

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

      6. metadata-eval 0.1%

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

      7. sub-neg 0.1%

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

      8. metadata-eval 0.1%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 74.7%

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

      11. sqr-neg 74.7%

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

      12. sqrt-unprod 74.7%

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

      13. add-sqr-sqrt 74.7%

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

   5. Applied egg-rr 74.7%

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

      1. *-commutative 74.7%

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

      2. distribute-rgt1-in 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in v around 0 74.7%

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

      1. associate-/l* 74.7%

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

      2. +-commutative 74.7%

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

   10. Applied egg-rr 74.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 + m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 1.2× speedup

Math

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

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 99.8%

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

6. Taylor expanded in v around 0 100.0%

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

7. Taylor expanded in m around inf 97.8%

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

8. Final simplification 97.8%

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

Alternative 8: 87.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 47.7%

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

   1. sub-neg 47.7%

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

   2. distribute-lft-in 47.7%

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

   3. *-commutative 47.7%

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

   4. *-un-lft-identity 47.7%

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

   5. sub-neg 47.7%

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

   6. metadata-eval 47.7%

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

   7. sub-neg 47.7%

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

   8. metadata-eval 47.7%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 86.5%

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

   11. sqr-neg 86.5%

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

   12. sqrt-unprod 86.5%

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

   13. add-sqr-sqrt 86.5%

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

5. Applied egg-rr 86.5%

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

   1. *-commutative 86.5%

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

   2. distribute-rgt1-in 86.5%

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

7. Simplified 86.5%

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

8. Final simplification 86.5%

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

Alternative 9: 27.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85 \cdot 10^{-34}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= m 2.85e-34) -1.0 m))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.85e-34) {
		tmp = -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 <= 2.85d-34) then
        tmp = -1.0d0
    else
        tmp = m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.85e-34) {
		tmp = -1.0;
	} else {
		tmp = m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.85e-34:
		tmp = -1.0
	else:
		tmp = m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.85e-34)
		tmp = -1.0;
	else
		tmp = m;
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.85e-34)
		tmp = -1.0;
	else
		tmp = m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.85e-34], -1.0, m]

Derivation

1. Split input into 2 regimes

2. if m < 2.84999999999999987e-34

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-/l* 99.8%

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

      4. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in m around 0 50.7%

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

   if 2.84999999999999987e-34 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf 91.2%

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

      1. neg-mul-1 91.2%

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

   5. Simplified 91.2%

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

   6. Taylor expanded in m around 0 5.5%

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

Alternative 10: 75.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{m}{v} + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in m around 0 47.7%

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

4. Taylor expanded in m around 0 72.1%

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

5. Final simplification 72.1%

   \[\leadsto \frac{m}{v} + -1 \]
6. Add Preprocessing

Alternative 11: 27.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ m + -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in v around inf 25.7%

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

   1. neg-mul-1 25.7%

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

   2. neg-sub0 25.7%

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

   3. associate--r- 25.7%

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

   4. metadata-eval 25.7%

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

7. Simplified 25.7%

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

8. Final simplification 25.7%

   \[\leadsto m + -1 \]
9. Add Preprocessing

Alternative 12: 25.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -1.0

Julia

function code(m, v)
	return -1.0
end

MATLAB

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

Wolfram

code[m_, v_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 23.2%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))