a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 40.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. associate-*r/ 99.8%

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

   2. clear-num 99.8%

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

6. Applied egg-rr 99.8%

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

   1. distribute-lft-in 99.8%

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

   2. un-div-inv 99.9%

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

   3. associate-/r* 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2e-159)
   (- m)
   (if (<= m 3.8e-14) (/ m (/ v m)) (* m (- -1.0 (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-159) {
		tmp = -m;
	} else if (m <= 3.8e-14) {
		tmp = m / (v / m);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.2d-159) then
        tmp = -m
    else if (m <= 3.8d-14) then
        tmp = m / (v / m)
    else
        tmp = m * ((-1.0d0) - (m / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-159) {
		tmp = -m;
	} else if (m <= 3.8e-14) {
		tmp = m / (v / m);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.2e-159:
		tmp = -m
	elif m <= 3.8e-14:
		tmp = m / (v / m)
	else:
		tmp = m * (-1.0 - (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.2e-159)
		tmp = Float64(-m);
	elseif (m <= 3.8e-14)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(-1.0 - Float64(m / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.2e-159)
		tmp = -m;
	elseif (m <= 3.8e-14)
		tmp = m / (v / m);
	else
		tmp = m * (-1.0 - (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.2e-159], (-m), If[LessEqual[m, 3.8e-14], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * N[(-1.0 - N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.2e-159

   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 80.8%

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

      1. neg-mul-1 80.8%

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

   7. Simplified 80.8%

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

   if 2.2e-159 < m < 3.8000000000000002e-14

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-/l* 99.5%

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

      4. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in m around inf 49.4%

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   6. Taylor expanded in m around 0 78.3%

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

      1. unpow2 78.3%

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

      2. associate-*r/ 79.6%

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

      3. clear-num 79.4%

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

      4. associate-/r/ 79.4%

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

      5. associate-*r* 79.4%

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

      6. div-inv 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r/ 79.7%

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

   10. Applied egg-rr 79.7%

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

   if 3.8000000000000002e-14 < 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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   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 1.1%

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

      1. clear-num 1.1%

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

      2. frac-2neg 1.1%

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

      3. pow1 1.1%

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

      4. metadata-eval 1.1%

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

      5. sqrt-pow1 76.3%

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

      6. pow2 76.3%

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

      7. sqr-neg 76.3%

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

      8. sqrt-prod 76.3%

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

      9. add-sqr-sqrt 76.3%

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

      10. div-inv 76.3%

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

   9. Applied egg-rr 76.3%

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

   10. Taylor expanded in m around 0 76.3%

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

       1. neg-mul-1 76.3%

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

       2. sub-neg 76.3%

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

       3. metadata-eval 76.3%

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

       4. +-commutative 76.3%

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

       5. sub-neg 76.3%

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

   12. Simplified 76.3%

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

Alternative 3: 37.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.9 \cdot 10^{-159} \lor \neg \left(m \leq 3.8 \cdot 10^{-14}\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 1.9e-159) (not (<= m 3.8e-14))) (- m) (/ m (/ v m))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 1.9e-159) || !(m <= 3.8e-14)) {
		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 <= 1.9d-159) .or. (.not. (m <= 3.8d-14))) 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 <= 1.9e-159) || !(m <= 3.8e-14)) {
		tmp = -m;
	} else {
		tmp = m / (v / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 1.9e-159) or not (m <= 3.8e-14):
		tmp = -m
	else:
		tmp = m / (v / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 1.9e-159) || !(m <= 3.8e-14))
		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 <= 1.9e-159) || ~((m <= 3.8e-14)))
		tmp = -m;
	else
		tmp = m / (v / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 1.9e-159], N[Not[LessEqual[m, 3.8e-14]], $MachinePrecision]], (-m), N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.9e-159 or 3.8000000000000002e-14 < 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 34.4%

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

      1. neg-mul-1 34.4%

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

   7. Simplified 34.4%

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

   if 1.9e-159 < m < 3.8000000000000002e-14

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-/l* 99.5%

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

      4. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in m around inf 49.4%

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   6. Taylor expanded in m around 0 78.3%

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

      1. unpow2 78.3%

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

      2. associate-*r/ 79.6%

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

      3. clear-num 79.4%

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

      4. associate-/r/ 79.4%

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

      5. associate-*r* 79.4%

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

      6. div-inv 79.6%

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

   8. Applied egg-rr 79.6%

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

      1. associate-/r/ 79.7%

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

   10. Applied egg-rr 79.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.9 \cdot 10^{-159} \lor \neg \left(m \leq 3.8 \cdot 10^{-14}\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-160} \lor \neg \left(m \leq 3.8 \cdot 10^{-14}\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (or (<= m 8.5e-160) (not (<= m 3.8e-14))) (- m) (* m (/ m v))))

C

double code(double m, double v) {
	double tmp;
	if ((m <= 8.5e-160) || !(m <= 3.8e-14)) {
		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 <= 8.5d-160) .or. (.not. (m <= 3.8d-14))) 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 <= 8.5e-160) || !(m <= 3.8e-14)) {
		tmp = -m;
	} else {
		tmp = m * (m / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if (m <= 8.5e-160) or not (m <= 3.8e-14):
		tmp = -m
	else:
		tmp = m * (m / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if ((m <= 8.5e-160) || !(m <= 3.8e-14))
		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 <= 8.5e-160) || ~((m <= 3.8e-14)))
		tmp = -m;
	else
		tmp = m * (m / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[Or[LessEqual[m, 8.5e-160], N[Not[LessEqual[m, 3.8e-14]], $MachinePrecision]], (-m), N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.49999999999999959e-160 or 3.8000000000000002e-14 < 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 34.4%

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

      1. neg-mul-1 34.4%

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

   7. Simplified 34.4%

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

   if 8.49999999999999959e-160 < m < 3.8000000000000002e-14

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-/l* 99.5%

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

      4. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in m around inf 49.4%

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]

   6. Taylor expanded in m around 0 78.3%

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

      1. unpow2 78.3%

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

      2. associate-*r/ 79.6%

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

      3. clear-num 79.4%

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

      4. associate-/r/ 79.4%

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

      5. associate-*r* 79.4%

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

      6. div-inv 79.6%

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

   8. Applied egg-rr 79.6%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-160} \lor \neg \left(m \leq 3.8 \cdot 10^{-14}\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. un-div-inv 99.8%

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

      3. associate-/r* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in m around 0 99.2%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   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. pow1 0.1%

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

      4. metadata-eval 0.1%

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

      5. sqrt-pow1 77.3%

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

      6. pow2 77.3%

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

      7. sqr-neg 77.3%

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

      8. sqrt-prod 77.3%

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

      9. add-sqr-sqrt 77.3%

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

      10. div-inv 77.3%

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

   9. Applied egg-rr 77.3%

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

   10. Taylor expanded in v around 0 77.3%

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

       1. +-commutative 77.3%

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

       2. mul-1-neg 77.3%

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

       3. unsub-neg 77.3%

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

       4. mul-1-neg 77.3%

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

       5. unpow2 77.3%

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

       6. distribute-rgt-neg-in 77.3%

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

       7. mul-1-neg 77.3%

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

       8. distribute-lft-out-- 77.3%

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

       9. mul-1-neg 77.3%

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

   12. Simplified 77.3%

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

4. Final simplification 88.9%

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

Alternative 6: 88.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.2%

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

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   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. pow1 0.1%

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

      4. metadata-eval 0.1%

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

      5. sqrt-pow1 77.3%

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

      6. pow2 77.3%

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

      7. sqr-neg 77.3%

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

      8. sqrt-prod 77.3%

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

      9. add-sqr-sqrt 77.3%

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

      10. div-inv 77.3%

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

   9. Applied egg-rr 77.3%

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

   10. Taylor expanded in v around 0 77.3%

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

       1. +-commutative 77.3%

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

       2. mul-1-neg 77.3%

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

       3. unsub-neg 77.3%

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

       4. mul-1-neg 77.3%

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

       5. unpow2 77.3%

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

       6. distribute-rgt-neg-in 77.3%

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

       7. mul-1-neg 77.3%

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

       8. distribute-lft-out-- 77.3%

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

       9. mul-1-neg 77.3%

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

   12. Simplified 77.3%

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

4. Final simplification 88.9%

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

Alternative 7: 88.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-/l* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in m around 0 99.2%

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

   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* 100.0%

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

      4. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v} + -1\right)} \]
   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. pow1 0.1%

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

      4. metadata-eval 0.1%

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

      5. sqrt-pow1 77.3%

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

      6. pow2 77.3%

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

      7. sqr-neg 77.3%

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

      8. sqrt-prod 77.3%

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

      9. add-sqr-sqrt 77.3%

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

      10. div-inv 77.3%

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

   9. Applied egg-rr 77.3%

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

   10. Taylor expanded in m around 0 77.3%

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

       1. neg-mul-1 77.3%

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

       2. sub-neg 77.3%

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

       3. metadata-eval 77.3%

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

       4. +-commutative 77.3%

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

       5. sub-neg 77.3%

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

   12. Simplified 77.3%

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

4. Final simplification 88.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 27.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-/l* 99.8%

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

   4. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in m around 0 30.4%

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

   1. neg-mul-1 30.4%

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

7. Simplified 30.4%

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

Reproduce

herbie shell --seed 2024106 
(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))