a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 10

Speedup: N/A×

• 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*r/ 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-145}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(-m\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55e-145) (- m) (if (<= m 1.0) (/ m (/ v m)) (* (/ m v) (- m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-145) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m / 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 <= 2.55d-145) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = (m / v) * -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-145) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m / v) * -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55e-145:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = (m / v) * -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55e-145)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(Float64(m / v) * Float64(-m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55e-145)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = (m / v) * -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55e-145], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(N[(m / v), $MachinePrecision] * (-m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.54999999999999992e-145

   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 76.6%

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

      1. neg-mul-1 76.6%

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

   7. Simplified 76.6%

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

   if 2.54999999999999992e-145 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 76.3%

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

      1. associate-/l* 76.1%

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

   7. Simplified 76.1%

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

      1. *-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. associate-*r* 76.1%

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

      4. associate-/r/ 76.1%

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

      5. *-commutative 76.1%

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

      6. clear-num 76.1%

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

      7. un-div-inv 76.3%

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

   9. Applied egg-rr 76.3%

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

   10. Taylor expanded in m around 0 72.7%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-/r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. clear-num 99.9%

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

      7. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in m around 0 0.1%

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

       1. frac-2neg 0.1%

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

       2. mul-1-neg 0.1%

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

       3. associate-/r/ 0.1%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 85.1%

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

       6. mul-1-neg 85.1%

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

       7. mul-1-neg 85.1%

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

       8. sqr-neg 85.1%

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

       9. sqrt-unprod 82.4%

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

       10. add-sqr-sqrt 82.4%

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

   12. Applied egg-rr 82.4%

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

Alternative 3: 38.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-142} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 2.95e-142) (not (<= m 1.0))) (- m) (/ m (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 2.95e-142) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m / (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 <= 2.95d-142) .or. (.not. (m <= 1.0d0))) then
        tmp = -m
    else
        tmp = m / (v / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if ((m <= 2.95e-142) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 2.95e-142) or not (m <= 1.0):
		tmp = -m
	else:
		tmp = m / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 2.95e-142) || !(m <= 1.0))
		tmp = Float64(-m);
	else
		tmp = Float64(m / Float64(v / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if ((m <= 2.95e-142) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 2.95e-142], N[Not[LessEqual[m, 1.0]], $MachinePrecision]], (-m), N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.94999999999999983e-142 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 29.6%

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

      1. neg-mul-1 29.6%

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

   7. Simplified 29.6%

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

   if 2.94999999999999983e-142 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 76.3%

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

      1. associate-/l* 76.1%

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

   7. Simplified 76.1%

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

      1. *-commutative 76.1%

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

      2. unpow2 76.1%

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

      3. associate-*r* 76.1%

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

      4. associate-/r/ 76.1%

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

      5. *-commutative 76.1%

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

      6. clear-num 76.1%

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

      7. un-div-inv 76.3%

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

   9. Applied egg-rr 76.3%

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

   10. Taylor expanded in m around 0 72.7%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-142} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-166} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 1.35e-166) (not (<= m 1.0))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 1.35e-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 <= 1.35d-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 <= 1.35e-166) || !(m <= 1.0)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 1.35e-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 <= 1.35e-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 <= 1.35e-166) || ~((m <= 1.0)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 1.35e-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 < 1.35000000000000003e-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 28.9%

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

      1. neg-mul-1 28.9%

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

   7. Simplified 28.9%

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

   if 1.35000000000000003e-166 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/l* 99.6%

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

      4. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in v around 0 72.6%

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

      1. associate-/l* 72.4%

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

   7. Simplified 72.4%

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

      1. *-commutative 72.4%

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

      2. unpow2 72.4%

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

      3. associate-*r* 74.2%

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

      4. associate-/r/ 74.2%

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

      5. *-commutative 74.2%

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

      6. clear-num 74.1%

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

      7. un-div-inv 74.3%

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

   9. Applied egg-rr 74.3%

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

   10. Taylor expanded in m around 0 71.0%

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

       1. associate-/r/ 71.0%

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

   12. Applied egg-rr 71.0%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-166} \lor \neg \left(m \leq 1\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.05e-24) (* m (/ (- m v) v)) (/ (* m (* m (- 1.0 m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.05e-24) {
		tmp = m * ((m - v) / v);
	} 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 <= 1.05d-24) then
        tmp = m * ((m - v) / v)
    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 <= 1.05e-24) {
		tmp = m * ((m - v) / v);
	} else {
		tmp = (m * (m * (1.0 - m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.05e-24:
		tmp = m * ((m - v) / v)
	else:
		tmp = (m * (m * (1.0 - m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.05e-24)
		tmp = Float64(m * Float64(Float64(m - v) / v));
	else
		tmp = Float64(Float64(m * Float64(m * Float64(1.0 - m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.05e-24)
		tmp = m * ((m - v) / v);
	else
		tmp = (m * (m * (1.0 - m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.05e-24], N[(m * N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.05e-24

   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. *-commutative 99.8%

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

      3. associate-/l* 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

   6. Taylor expanded in m around 0 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   9. Simplified 99.8%

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

   if 1.05e-24 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-/r/ 99.9%

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

      5. associate-/r/ 99.9%

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

      6. associate-*l/ 99.9%

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

      7. associate-*l/ 99.9%

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

      8. *-commutative 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-24}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-25}:\\ \;\;\;\;m \cdot \frac{m - v}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-25) (* m (/ (- m v) v)) (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-25) {
		tmp = m * ((m - v) / v);
	} 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 <= 5d-25) then
        tmp = m * ((m - v) / v)
    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 <= 5e-25) {
		tmp = m * ((m - v) / v);
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-25:
		tmp = m * ((m - v) / v)
	else:
		tmp = m * ((m * (1.0 - m)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-25)
		tmp = Float64(m * Float64(Float64(m - v) / v));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-25)
		tmp = m * ((m - v) / v);
	else
		tmp = m * ((m * (1.0 - m)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-25], N[(m * N[(N[(m - v), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.99999999999999962e-25

   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. *-commutative 99.8%

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

      3. associate-/l* 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

   6. Taylor expanded in m around 0 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   9. Simplified 99.8%

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

   if 4.99999999999999962e-25 < 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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in v around 0 99.9%

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

Alternative 7: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

   6. Taylor expanded in m around 0 97.9%

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

   7. Taylor expanded in v around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

   9. Simplified 97.9%

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

   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. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

   6. Taylor expanded in m around 0 0.1%

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

      1. clear-num 0.1%

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

      2. un-div-inv 0.1%

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

      3. frac-2neg 0.1%

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

      4. mul-1-neg 0.1%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 85.1%

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

      7. mul-1-neg 85.1%

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

      8. mul-1-neg 85.1%

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

      9. sqr-neg 85.1%

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

      10. sqrt-unprod 82.4%

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

      11. add-sqr-sqrt 82.4%

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

      12. distribute-neg-in 82.4%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 82.4%

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

      15. mul-1-neg 82.4%

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

      16. mul-1-neg 82.4%

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

      17. sqr-neg 82.4%

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

      18. sqrt-unprod 82.4%

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

      19. add-sqr-sqrt 82.4%

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

      20. mul-1-neg 82.4%

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

      21. sub-neg 82.4%

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

   8. Applied egg-rr 82.4%

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

Alternative 8: 87.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 99.8%

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

      4. fma-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 99.8%

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

   6. Taylor expanded in m around 0 97.9%

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

   7. Taylor expanded in v around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

   9. Simplified 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-/r/ 99.9%

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

      5. *-commutative 99.9%

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

      6. clear-num 99.9%

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

      7. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in m around 0 0.1%

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

       1. frac-2neg 0.1%

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

       2. mul-1-neg 0.1%

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

       3. associate-/r/ 0.1%

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

       4. add-sqr-sqrt 0.0%

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

       5. sqrt-unprod 85.1%

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

       6. mul-1-neg 85.1%

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

       7. mul-1-neg 85.1%

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

       8. sqr-neg 85.1%

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

       9. sqrt-unprod 82.4%

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

       10. add-sqr-sqrt 82.4%

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

   12. Applied egg-rr 82.4%

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

Alternative 9: 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 10: 26.7% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := (-m)

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 27.7%

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

   1. neg-mul-1 27.7%

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

7. Simplified 27.7%

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

Reproduce

herbie shell --seed 2024086 
(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))