a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 10

Speedup: 1.0×

• 42.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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* 99.8%

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

   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. clear-num 99.8%

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

   2. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 6.5e-26) (* (- (/ m v) 1.0) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 6.5e-26) {
		tmp = ((m / v) - 1.0) * 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 <= 6.5d-26) then
        tmp = ((m / v) - 1.0d0) * 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 <= 6.5e-26) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 6.5e-26:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 6.5e-26)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * 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 <= 6.5e-26)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 6.5e-26], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $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 < 6.5e-26

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

   if 6.5e-26 < 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. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-/l* 99.9%

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

      2. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

Alternative 3: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-37}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-37) (* (- (/ m v) 1.0) m) (* m (/ m (/ v (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-37) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = m * (m / (v / (1.0 - 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 <= 1d-37) then
        tmp = ((m / v) - 1.0d0) * m
    else
        tmp = m * (m / (v / (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1e-37) {
		tmp = ((m / v) - 1.0) * m;
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-37:
		tmp = ((m / v) - 1.0) * m
	else:
		tmp = m * (m / (v / (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-37)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(m * Float64(m / Float64(v / Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-37)
		tmp = ((m / v) - 1.0) * m;
	else
		tmp = m * (m / (v / (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-37], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(m * N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.00000000000000007e-37

   1. Initial program 99.8%

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

   3. Taylor expanded in m around 0 99.8%

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

   if 1.00000000000000007e-37 < 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. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. associate-/l* 99.9%

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

      2. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. clear-num 99.8%

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

       2. un-div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

Alternative 4: 87.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   3. Taylor expanded in m around 0 98.6%

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

   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. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. clear-num 0.1%

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

      2. frac-2neg 0.1%

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

      3. mul-1-neg 0.1%

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

      4. distribute-frac-neg 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 78.5%

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

      7. mul-1-neg 78.5%

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

      8. mul-1-neg 78.5%

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

      9. sqr-neg 78.5%

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

      10. sqrt-unprod 78.0%

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

      11. add-sqr-sqrt 78.0%

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

   9. Applied egg-rr 78.0%

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

Alternative 5: 51.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(m * -1.0), $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 98.6%

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

   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 5.7%

      \[\leadsto m \cdot \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 7: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8 \cdot 10^{-151}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \mathbf{else}:\\ \;\;\;\;m \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 8e-151) (* (/ m v) m) (* m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 8e-151) {
		tmp = (m / v) * m;
	} else {
		tmp = m * -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 8d-151) then
        tmp = (m / v) * m
    else
        tmp = m * (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 8e-151) {
		tmp = (m / v) * m;
	} else {
		tmp = m * -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 8e-151:
		tmp = (m / v) * m
	else:
		tmp = m * -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 8e-151)
		tmp = Float64(Float64(m / v) * m);
	else
		tmp = Float64(m * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 8e-151)
		tmp = (m / v) * m;
	else
		tmp = m * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 8e-151], N[(N[(m / v), $MachinePrecision] * m), $MachinePrecision], N[(m * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 7.9999999999999995e-151

   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 48.1%

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

   4. Taylor expanded in m around inf 36.7%

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

   if 7.9999999999999995e-151 < v

   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 39.1%

      \[\leadsto m \cdot \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 37.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot -1\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= v 8.6e-151) (/ m (/ v m)) (* m -1.0)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 8.6e-151) {
		tmp = m / (v / m);
	} else {
		tmp = m * -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 8.6d-151) then
        tmp = m / (v / m)
    else
        tmp = m * (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (v <= 8.6e-151) {
		tmp = m / (v / m);
	} else {
		tmp = m * -1.0;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 8.6e-151:
		tmp = m / (v / m)
	else:
		tmp = m * -1.0
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 8.6e-151)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (v <= 8.6e-151)
		tmp = m / (v / m);
	else
		tmp = m * -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 8.6e-151], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 8.60000000000000035e-151

   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 48.1%

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

   4. Taylor expanded in m around inf 36.7%

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

      1. *-commutative 36.7%

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

      2. clear-num 36.6%

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

      3. un-div-inv 36.7%

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

   6. Applied egg-rr 36.7%

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

   if 8.60000000000000035e-151 < v

   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 39.1%

      \[\leadsto m \cdot \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 26.6% accurate, 3.7× speedup

Math

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

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 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 26.0%

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

Alternative 10: 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. Add Preprocessing

3. Taylor expanded in v around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. clear-num 99.8%

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

   3. un-div-inv 99.8%

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

   4. +-commutative 99.8%

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

   5. fma-define 99.8%

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

   6. add-sqr-sqrt 0.0%

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

   7. sqrt-unprod 76.1%

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

   8. mul-1-neg 76.1%

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

   9. mul-1-neg 76.1%

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

   10. sqr-neg 76.1%

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

   11. sqrt-unprod 76.1%

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

   12. add-sqr-sqrt 76.1%

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

5. Applied egg-rr 76.1%

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

6. Taylor expanded in m around 0 3.0%

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

Reproduce

herbie shell --seed 2024052 -o generate:simplify
(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))