a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.9%

Time: 8.1s

Alternatives: 12

Speedup: 1.0×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. metadata-eval N/A

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

   5. neg-mul-1 N/A

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

   6. unsub-neg N/A

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

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v} \cdot m\right), \color{blue}{m}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v} \cdot m\right), m\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot \frac{m}{v}\right) \cdot m\right), m\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(1 - m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(\frac{m}{v}\right), m\right)\right), m\right) \]

   14. /-lowering-/.f64 99.8%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right), m\right) \]

4. Applied egg-rr 99.8%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\right), m\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{m}{v}\right), m\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1}{\frac{v}{m}}\right), m\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{\frac{v}{m}}\right), m\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - m\right) \cdot m\right), \left(\frac{v}{m}\right)\right), m\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

   9. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \mathsf{/.f64}\left(v, m\right)\right), m\right) \]

6. Applied egg-rr 99.8%

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

Alternative 2: 61.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{-1}{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6e-166)
   (- 0.0 m)
   (if (<= m 1.0) (/ m (/ v m)) (* (* m m) (/ -1.0 m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m * m) * (-1.0 / m);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d-166) then
        tmp = 0.0d0 - m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = (m * m) * ((-1.0d0) / m)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m * m) * (-1.0 / m);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6e-166:
		tmp = 0.0 - m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = (m * m) * (-1.0 / m)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6e-166)
		tmp = Float64(0.0 - m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(Float64(m * m) * Float64(-1.0 / m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6e-166)
		tmp = 0.0 - m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = (m * m) * (-1.0 / m);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6e-166], N[(0.0 - m), $MachinePrecision], If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] * N[(-1.0 / m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.59999999999999989e-166

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 81.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 81.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 81.7%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 81.7%

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

   if 2.59999999999999989e-166 < m < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, m\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), m\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), m\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), m\right) \]

      15. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), m\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), m\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), v\right), m\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), v\right), m\right) \]

      19. --lowering--.f64 77.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), v\right), m\right) \]

   5. Simplified 77.4%

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

   6. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({m}^{2}\right), \color{blue}{v}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(m \cdot m\right), v\right) \]

      3. *-lowering-*.f64 71.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, m\right), v\right) \]

   8. Simplified 71.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(m, \color{blue}{\left(\frac{v}{m}\right)}\right) \]

      5. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \color{blue}{m}\right)\right) \]

   10. Applied egg-rr 73.7%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 5.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 5.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 5.7%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 5.7%

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

      1. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + m\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 \cdot 0 - m \cdot m}{0 - m}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 - m \cdot m}{0 - m}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{0 - m}\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{\mathsf{neg}\left(m\right)}\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(m \cdot m\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(m\right)}} \]

      7. div-inv N/A

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

      8. remove-double-neg N/A

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

      9. neg-sub0 N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot m\right), \color{blue}{\left(\frac{1}{0 - m}\right)}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\color{blue}{1}}{0 - m}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{0} - m}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(m\right)}\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{-1}{\color{blue}{m}}\right)\right) \]

      15. /-lowering-/.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \mathsf{/.f64}\left(-1, \color{blue}{m}\right)\right) \]

   9. Applied egg-rr 56.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{-1}{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.55e-166) (- 0.0 m) (if (<= m 1.0) (/ m (/ v m)) (- 0.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.55d-166) then
        tmp = 0.0d0 - m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = 0.0d0 - m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.55e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.55e-166:
		tmp = 0.0 - m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = 0.0 - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.55e-166)
		tmp = Float64(0.0 - m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(0.0 - m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.55e-166)
		tmp = 0.0 - m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = 0.0 - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.55e-166], N[(0.0 - m), $MachinePrecision], If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(0.0 - m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5500000000000001e-166 or 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 32.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 32.9%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 32.9%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 32.9%

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

   if 2.5500000000000001e-166 < m < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, m\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), m\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), m\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), m\right) \]

      15. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), m\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), m\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), v\right), m\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), v\right), m\right) \]

      19. --lowering--.f64 77.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), v\right), m\right) \]

   5. Simplified 77.4%

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

   6. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({m}^{2}\right), \color{blue}{v}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(m \cdot m\right), v\right) \]

      3. *-lowering-*.f64 71.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(m, m\right), v\right) \]

   8. Simplified 71.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(m, \color{blue}{\left(\frac{v}{m}\right)}\right) \]

      5. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \color{blue}{m}\right)\right) \]

   10. Applied egg-rr 73.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.55 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.6e-166) (- 0.0 m) (if (<= m 1.0) (* m (/ m v)) (- 0.0 m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.6d-166) then
        tmp = 0.0d0 - m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = 0.0d0 - m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.6e-166) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.6e-166:
		tmp = 0.0 - m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = 0.0 - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.6e-166)
		tmp = Float64(0.0 - m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(0.0 - m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.6e-166)
		tmp = 0.0 - m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = 0.0 - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.6e-166], N[(0.0 - m), $MachinePrecision], If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(0.0 - m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 2.59999999999999989e-166 or 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 32.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 32.9%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 32.9%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 32.9%

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

   if 2.59999999999999989e-166 < m < 1

   1. Initial program 99.5%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, m\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), m\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), m\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), m\right) \]

      15. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), m\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), m\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), v\right), m\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), v\right), m\right) \]

      19. --lowering--.f64 77.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), v\right), m\right) \]

   5. Simplified 77.4%

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

   6. Taylor expanded in m around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, m\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right) \]

   8. Simplified 73.6%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-166}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-12) (- (/ m (/ v m)) m) (* m (/ m (/ v (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5d-12) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * (m / (v / (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-12:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m / (v / (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(m / Float64(v / Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m / (v / (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-12], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v} \cdot m\right), \color{blue}{m}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v} \cdot m\right), m\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot \frac{m}{v}\right) \cdot m\right), m\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(1 - m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(\frac{m}{v}\right), m\right)\right), m\right) \]

      14. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right), m\right) \]

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\right), m\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{m}{v}\right), m\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1}{\frac{v}{m}}\right), m\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{\frac{v}{m}}\right), m\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - m\right) \cdot m\right), \left(\frac{v}{m}\right)\right), m\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \mathsf{/.f64}\left(v, m\right)\right), m\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{m}, \mathsf{/.f64}\left(v, m\right)\right), m\right) \]
   8. Step-by-step derivation

   9. Simplified 99.4%

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

   if 4.9999999999999997e-12 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, m\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), m\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), m\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), m\right) \]

      15. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), m\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), m\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), v\right), m\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), v\right), m\right) \]

      19. --lowering--.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), v\right), m\right) \]

   5. Simplified 99.9%

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

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} \cdot \left(1 - m\right)\right), m\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), m\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), m\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), m\right) \]

      5. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), m\right) \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-12) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 5d-12) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 5e-12) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5e-12:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5e-12)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-12)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5e-12], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.9999999999999997e-12

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v} \cdot m\right), \color{blue}{m}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v} \cdot m\right), m\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot \frac{m}{v}\right) \cdot m\right), m\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(1 - m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(\frac{m}{v}\right), m\right)\right), m\right) \]

      14. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right), m\right) \]

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\right), m\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{m}{v}\right), m\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1}{\frac{v}{m}}\right), m\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{\frac{v}{m}}\right), m\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - m\right) \cdot m\right), \left(\frac{v}{m}\right)\right), m\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \mathsf{/.f64}\left(v, m\right)\right), m\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{m}, \mathsf{/.f64}\left(v, m\right)\right), m\right) \]
   8. Step-by-step derivation

   9. Simplified 99.4%

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

   if 4.9999999999999997e-12 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({m}^{2} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)\right)}, m\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{1}{m \cdot v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left({m}^{2} \cdot \frac{\frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{m}^{2} \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(m \cdot m\right) \cdot \frac{1}{m}}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(m \cdot \frac{1}{m}\right)}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      6. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      7. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot 1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - {m}^{2} \cdot \frac{1}{v}\right), m\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - \left(m \cdot m\right) \cdot \frac{1}{v}\right), m\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \left(m \cdot \frac{1}{v}\right)\right), m\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m \cdot 1}{v}\right), m\right) \]

      13. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1}{v} - m \cdot \frac{m}{v}\right), m\right) \]

      14. distribute-lft-out-- N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \left(\frac{1}{v} - \frac{m}{v}\right)\right), m\right) \]

      15. div-sub N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), m\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), m\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), v\right), m\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), v\right), m\right) \]

      19. --lowering--.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), v\right), m\right) \]

   5. Simplified 99.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v}\right), m\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 - m}{v} \cdot m\right), m\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1 - m}{v}\right), m\right), m\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - m\right), v\right), m\right), m\right) \]

      5. --lowering--.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, m\right), v\right), m\right), m\right) \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - m \cdot m}{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0) (- (/ m (/ v m)) m) (/ (- 0.0 (* m m)) m)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.0) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (0.0 - (m * m)) / 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 / (v / m)) - m
    else
        tmp = (0.0d0 - (m * m)) / m
    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 = (0.0 - (m * m)) / m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m / (v / m)) - m
	else:
		tmp = (0.0 - (m * m)) / m
	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(0.0 - Float64(m * m)) / m);
	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 = (0.0 - (m * m)) / m;
	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[(0.0 - N[(m * m), $MachinePrecision]), $MachinePrecision] / m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v} \cdot m\right), \color{blue}{m}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v} \cdot m\right), m\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot \frac{m}{v}\right) \cdot m\right), m\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(1 - m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(\frac{m}{v}\right), m\right)\right), m\right) \]

      14. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right), m\right) \]

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\right), m\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{m}{v}\right), m\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1}{\frac{v}{m}}\right), m\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{\frac{v}{m}}\right), m\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - m\right) \cdot m\right), \left(\frac{v}{m}\right)\right), m\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \mathsf{/.f64}\left(v, m\right)\right), m\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{m}, \mathsf{/.f64}\left(v, m\right)\right), m\right) \]
   8. Step-by-step derivation

   9. Simplified 97.7%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 5.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 5.7%

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

      1. flip-- N/A

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

      2. +-lft-identity N/A

         \[\leadsto \frac{0 \cdot 0 - m \cdot m}{m} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 \cdot 0 - m \cdot m\right), \color{blue}{m}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - m \cdot m\right), m\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(m \cdot m\right)\right), m\right) \]

      6. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(m, m\right)\right), m\right) \]

   7. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\frac{0 - m \cdot m}{m}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * m) * (-1.0 / m)
	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 * m) * Float64(-1.0 / m));
	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) * (-1.0 / m);
	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 * m), $MachinePrecision] * N[(-1.0 / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. neg-mul-1 N/A

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

      6. unsub-neg N/A

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

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v} \cdot m\right), \color{blue}{m}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{v} \cdot m\right), m\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot \frac{m}{v}\right) \cdot m\right), m\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(1 - m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \left(\frac{m}{v} \cdot m\right)\right), m\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\left(\frac{m}{v}\right), m\right)\right), m\right) \]

      14. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, m\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(m, v\right), m\right)\right), m\right) \]

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\right), m\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{m}{v}\right), m\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\left(1 - m\right) \cdot m\right) \cdot \frac{1}{\frac{v}{m}}\right), m\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(1 - m\right) \cdot m}{\frac{v}{m}}\right), m\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - m\right) \cdot m\right), \left(\frac{v}{m}\right)\right), m\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m \cdot \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \left(1 - m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \left(\frac{v}{m}\right)\right), m\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(m, \mathsf{\_.f64}\left(1, m\right)\right), \mathsf{/.f64}\left(v, m\right)\right), m\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in m around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{m}, \mathsf{/.f64}\left(v, m\right)\right), m\right) \]
   8. Step-by-step derivation

   9. Simplified 97.7%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 5.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 5.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 5.7%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 5.7%

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

      1. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + m\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 \cdot 0 - m \cdot m}{0 - m}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 - m \cdot m}{0 - m}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{0 - m}\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{\mathsf{neg}\left(m\right)}\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(m \cdot m\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(m\right)}} \]

      7. div-inv N/A

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

      8. remove-double-neg N/A

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

      9. neg-sub0 N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot m\right), \color{blue}{\left(\frac{1}{0 - m}\right)}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\color{blue}{1}}{0 - m}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{0} - m}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(m\right)}\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{-1}{\color{blue}{m}}\right)\right) \]

      15. /-lowering-/.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \mathsf{/.f64}\left(-1, \color{blue}{m}\right)\right) \]

   9. Applied egg-rr 56.8%

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

Alternative 9: 74.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.7%

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

   3. Taylor expanded in m around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{m}{v}\right)}, 1\right), m\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 97.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, v\right), 1\right), m\right) \]

   5. Simplified 97.7%

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

   if 1 < m

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 5.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

   5. Simplified 5.7%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(m\right) \]

      2. neg-lowering-neg.f64 5.7%

         \[\leadsto \mathsf{neg.f64}\left(m\right) \]

   7. Applied egg-rr 5.7%

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

      1. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + m\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 \cdot 0 - m \cdot m}{0 - m}\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 - m \cdot m}{0 - m}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{0 - m}\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\mathsf{neg}\left(m \cdot m\right)}{\mathsf{neg}\left(m\right)}\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(m \cdot m\right)\right)\right)}{\color{blue}{\mathsf{neg}\left(m\right)}} \]

      7. div-inv N/A

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

      8. remove-double-neg N/A

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

      9. neg-sub0 N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(m \cdot m\right), \color{blue}{\left(\frac{1}{0 - m}\right)}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\color{blue}{1}}{0 - m}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\color{blue}{0} - m}\right)\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(m\right)}\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \left(\frac{-1}{\color{blue}{m}}\right)\right) \]

      15. /-lowering-/.f64 56.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(m, m\right), \mathsf{/.f64}\left(-1, \color{blue}{m}\right)\right) \]

   9. Applied egg-rr 56.8%

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

4. Final simplification 78.2%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m \cdot \left(1 - m\right)}{v}\right), 1\right), m\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), m\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1}{\frac{v}{1 - m}}\right), 1\right), m\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{m}{\frac{v}{1 - m}}\right), 1\right), m\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\frac{v}{1 - m}\right)\right), 1\right), m\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \left(1 - m\right)\right)\right), 1\right), m\right) \]

   7. --lowering--.f64 99.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \mathsf{/.f64}\left(v, \mathsf{\_.f64}\left(1, m\right)\right)\right), 1\right), m\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(m \cdot \frac{1 - m}{v}\right), 1\right), m\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1 - m}{v} \cdot m\right), 1\right), m\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{1 - m}{v}\right), m\right), 1\right), m\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 - m\right), v\right), m\right), 1\right), m\right) \]

   5. --lowering--.f64 99.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, m\right), v\right), m\right), 1\right), m\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 12: 27.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ 0 - m \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (- 0.0 m))

C

double code(double m, double v) {
	return 0.0 - m;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = 0.0d0 - m
end function

Java

public static double code(double m, double v) {
	return 0.0 - m;
}

Python

def code(m, v):
	return 0.0 - m

Julia

function code(m, v)
	return Float64(0.0 - m)
end

MATLAB

function tmp = code(m, v)
	tmp = 0.0 - m;
end

Wolfram

code[m_, v_] := N[(0.0 - m), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in m around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(m\right) \]

   2. neg-sub0 N/A

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

   3. --lowering--.f64 29.2%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{m}\right) \]

5. Simplified 29.2%

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

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(m\right) \]

   2. neg-lowering-neg.f64 29.2%

      \[\leadsto \mathsf{neg.f64}\left(m\right) \]

7. Applied egg-rr 29.2%

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

8. Final simplification 29.2%

   \[\leadsto 0 - m \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (m v)
  :name "a parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) m))