a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 12

Speedup: 1.0×

• 41.6% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-11) (- (/ m (+ v (/ v m))) m) (* m (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-11) {
		tmp = (m / (v + (v / m))) - m;
	} else {
		tmp = m * ((1.0 - 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 <= 1.6d-11) then
        tmp = (m / (v + (v / m))) - m
    else
        tmp = m * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-11) {
		tmp = (m / (v + (v / m))) - m;
	} else {
		tmp = m * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-11:
		tmp = (m / (v + (v / m))) - m
	else:
		tmp = m * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-11)
		tmp = Float64(Float64(m / Float64(v + Float64(v / m))) - m);
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-11)
		tmp = (m / (v + (v / m))) - m;
	else
		tmp = m * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-11], N[(N[(m / N[(v + N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.59999999999999997e-11

   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. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

      14. --lowering--.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. div-sub N/A

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

      7. *-rgt-identity N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. distribute-rgt-out-- N/A

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

      11. *-lft-identity N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. neg-sub0 N/A

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

      18. associate--r- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(m, \left(\left(v - 0\right) + \frac{v}{m}\right)\right), m\right) \]

      19. --rgt-identity N/A

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

      20. +-lowering-+.f64 N/A

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

      21. /-lowering-/.f64 99.9%

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

   7. Simplified 99.9%

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

   if 1.59999999999999997e-11 < m

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-lft-neg-in N/A

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

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

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

      7. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - m \cdot m\right), m\right), v\right), 1\right), m\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \left(m \cdot m\right)\right), m\right), v\right), 1\right), m\right) \]

      9. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(m, m\right)\right), m\right), v\right), 1\right), m\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{\color{blue}{\left(0 - m \cdot m\right) + m}}{v} - 1\right) \cdot m \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m + \left(0 - m \cdot m\right)\right), v\right), 1\right), m\right) \]

      2. sub0-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. *-lowering-*.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   9. Simplified 99.8%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. --lowering--.f64 N/A

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

       10. /-lowering-/.f64 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.15e-122)
   (- 0.0 m)
   (if (<= m 1.0) (/ m (/ v m)) (* m (- -1.0 (/ m v))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.15e-122) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		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 <= 1.15d-122) then
        tmp = 0.0d0 - m
    else if (m <= 1.0d0) 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 <= 1.15e-122) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (-1.0 - (m / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.15e-122:
		tmp = 0.0 - m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m * (-1.0 - (m / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.15e-122)
		tmp = Float64(0.0 - m);
	elseif (m <= 1.0)
		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 <= 1.15e-122)
		tmp = 0.0 - m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m * (-1.0 - (m / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.15e-122], N[(0.0 - m), $MachinePrecision], If[LessEqual[m, 1.0], 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 < 1.15000000000000003e-122

   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 73.8%

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

   5. Simplified 73.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 73.8%

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

   7. Applied egg-rr 73.8%

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

   if 1.15000000000000003e-122 < m < 1

   1. Initial program 99.6%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 91.1%

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

   5. Simplified 91.1%

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

   6. Taylor expanded in m around inf

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

      1. /-lowering-/.f64 75.7%

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

   8. Simplified 75.7%

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

      1. *-commutative 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 75.8%

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

   10. Applied egg-rr 75.8%

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

   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

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

      1. /-lowering-/.f64 0.1%

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

   5. Simplified 0.1%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. sub0-neg N/A

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

   7. Applied egg-rr 75.6%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. unsub-neg N/A

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

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

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

      6. /-lowering-/.f64 75.6%

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

   9. Applied egg-rr 75.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(-1 - \frac{m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2e-20) (- (/ m (/ v m)) m) (* m (/ (- 1.0 m) (/ v m)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-20) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((1.0 - 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 <= 2d-20) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * ((1.0d0 - m) / (v / m))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2e-20) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * ((1.0 - m) / (v / m));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-20:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * ((1.0 - m) / (v / m))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2e-20)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(Float64(1.0 - m) / Float64(v / m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2e-20)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * ((1.0 - m) / (v / m));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2e-20], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(1.0 - m), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.99999999999999989e-20

   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. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

      14. --lowering--.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 99.9%

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

   7. Simplified 99.9%

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

   if 1.99999999999999989e-20 < m

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-lft-neg-in N/A

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

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

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

      7. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - m \cdot m\right), m\right), v\right), 1\right), m\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \left(m \cdot m\right)\right), m\right), v\right), 1\right), m\right) \]

      9. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(m, m\right)\right), m\right), v\right), 1\right), m\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{\color{blue}{\left(0 - m \cdot m\right) + m}}{v} - 1\right) \cdot m \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m + \left(0 - m \cdot m\right)\right), v\right), 1\right), m\right) \]

      2. sub0-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. *-lowering-*.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in m around inf

      \[\leadsto \color{blue}{{m}^{3} \cdot \left(\frac{1}{m \cdot v} - \frac{1}{v}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

      2. div-sub N/A

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

      3. associate-/l* N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-out-- N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

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

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

   9. Simplified 99.7%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. --lowering--.f64 N/A

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

       10. /-lowering-/.f64 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{1 - m}{\frac{v}{m}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2e-20) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2e-20) {
		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 <= 2d-20) 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 <= 2e-20) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2e-20:
		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 <= 2e-20)
		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 <= 2e-20)
		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, 2e-20], 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 < 1.99999999999999989e-20

   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. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

      14. --lowering--.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 99.9%

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

   7. Simplified 99.9%

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

   if 1.99999999999999989e-20 < 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. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. rgt-mult-inverse N/A

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

      10. div-sub N/A

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

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

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

      12. --lowering--.f64 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (-1.0 - (m / 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(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 / (v / m)) - m;
	else
		tmp = m * (-1.0 - (m / 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[(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. 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. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

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

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

      14. --lowering--.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in m around 0

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

      1. /-lowering-/.f64 96.2%

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

   7. Simplified 96.2%

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

   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

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

      1. /-lowering-/.f64 0.1%

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

   5. Simplified 0.1%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. sub0-neg N/A

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

   7. Applied egg-rr 75.6%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. unsub-neg N/A

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

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

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

      6. /-lowering-/.f64 75.6%

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

   9. Applied egg-rr 75.6%

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

4. Final simplification 86.3%

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

Alternative 6: 87.0% 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. 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 96.2%

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

   5. Simplified 96.2%

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

   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

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

      1. /-lowering-/.f64 0.1%

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

   5. Simplified 0.1%

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

      1. frac-2neg N/A

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

      2. distribute-frac-neg2 N/A

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

      3. sub0-neg N/A

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

   7. Applied egg-rr 75.6%

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. unsub-neg N/A

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

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

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

      6. /-lowering-/.f64 75.6%

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

   9. Applied egg-rr 75.6%

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

4. Final simplification 86.3%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. distribute-lft-neg-in N/A

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

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

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

   7. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(0 - m \cdot m\right), m\right), v\right), 1\right), m\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \left(m \cdot m\right)\right), m\right), v\right), 1\right), m\right) \]

   9. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(m, m\right)\right), m\right), v\right), 1\right), m\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(\frac{\color{blue}{\left(0 - m \cdot m\right) + m}}{v} - 1\right) \cdot m \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(m + \left(0 - m \cdot m\right)\right), v\right), 1\right), m\right) \]

   2. sub0-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. *-lowering-*.f64 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 9: 38.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 1.6e-162) (/ m (/ v m)) (- 0.0 m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.6e-162) {
		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 (v <= 1.6d-162) 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 (v <= 1.6e-162) {
		tmp = m / (v / m);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.6e-162:
		tmp = m / (v / m)
	else:
		tmp = 0.0 - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.6e-162)
		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 (v <= 1.6e-162)
		tmp = m / (v / m);
	else
		tmp = 0.0 - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.6e-162], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(0.0 - m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.59999999999999988e-162

   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

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

      1. /-lowering-/.f64 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in m around inf

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

      1. /-lowering-/.f64 40.7%

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

   8. Simplified 40.7%

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

      1. *-commutative 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 40.7%

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

   10. Applied egg-rr 40.7%

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

   if 1.59999999999999988e-162 < v

   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 41.4%

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

   5. Simplified 41.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 41.4%

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

   7. Applied egg-rr 41.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 7.5 \cdot 10^{-163}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (if (<= v 7.5e-163) (* m (/ m v)) (- 0.0 m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 7.5e-163) {
		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 (v <= 7.5d-163) 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 (v <= 7.5e-163) {
		tmp = m * (m / v);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 7.5e-163:
		tmp = m * (m / v)
	else:
		tmp = 0.0 - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 7.5e-163)
		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 (v <= 7.5e-163)
		tmp = m * (m / v);
	else
		tmp = 0.0 - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 7.5e-163], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(0.0 - m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 7.49999999999999996e-163

   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

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

      1. /-lowering-/.f64 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in m around inf

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

      1. /-lowering-/.f64 40.7%

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

   8. Simplified 40.7%

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

   if 7.49999999999999996e-163 < v

   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 41.4%

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

   5. Simplified 41.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 41.4%

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

   7. Applied egg-rr 41.4%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 7.5 \cdot 10^{-163}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.3% 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 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 27.8%

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

5. Simplified 27.8%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 27.8%

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

7. Applied egg-rr 27.8%

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

8. Final simplification 27.8%

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

Alternative 12: 3.0% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (m v) :precision binary64 m)

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return m

Julia

function code(m, v)
	return m
end

MATLAB

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

Wolfram

code[m_, v_] := m

Derivation

1. Initial program 99.9%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
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 27.8%

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

5. Simplified 27.8%

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

   1. flip-- N/A

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

7. Applied egg-rr 3.1%

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

Reproduce

herbie shell --seed 2024192 
(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))