a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.0s

Alternatives: 9

Speedup: 1.0×

• 42.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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. 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 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 50000:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{elif}\;m \leq 1.36 \cdot 10^{+153}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(0 - m\right)}{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 50000.0)
   (* m (+ (/ m v) -1.0))
   (if (<= m 1.36e+153) (/ m (/ v m)) (/ (* m (- 0.0 m)) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 50000.0) {
		tmp = m * ((m / v) + -1.0);
	} else if (m <= 1.36e+153) {
		tmp = m / (v / m);
	} else {
		tmp = (m * (0.0 - 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 <= 50000.0d0) then
        tmp = m * ((m / v) + (-1.0d0))
    else if (m <= 1.36d+153) then
        tmp = m / (v / m)
    else
        tmp = (m * (0.0d0 - m)) / m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 50000.0) {
		tmp = m * ((m / v) + -1.0);
	} else if (m <= 1.36e+153) {
		tmp = m / (v / m);
	} else {
		tmp = (m * (0.0 - m)) / m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 50000.0:
		tmp = m * ((m / v) + -1.0)
	elif m <= 1.36e+153:
		tmp = m / (v / m)
	else:
		tmp = (m * (0.0 - m)) / m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 50000.0)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	elseif (m <= 1.36e+153)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(Float64(m * Float64(0.0 - m)) / m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 50000.0)
		tmp = m * ((m / v) + -1.0);
	elseif (m <= 1.36e+153)
		tmp = m / (v / m);
	else
		tmp = (m * (0.0 - m)) / m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 50000.0], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.36e+153], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(0.0 - m), $MachinePrecision]), $MachinePrecision] / m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 5e4

   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 95.0%

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

   5. Simplified 95.0%

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

   if 5e4 < m < 1.36000000000000008e153

   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 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. 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 0.1%

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

   8. Simplified 0.1%

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

      1. associate-/r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 0.1%

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

   10. Applied egg-rr 0.1%

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

       1. remove-double-neg N/A

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

       2. neg-lowering-neg.f64 N/A

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

       3. neg-sub0 N/A

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

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

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

       5. /-lowering-/.f64 15.8%

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

   12. Applied egg-rr 15.8%

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

   if 1.36000000000000008e153 < m

   1. Initial program 100.0%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   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 7.1%

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

   5. Simplified 7.1%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. sub0-neg N/A

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

      4. +-lft-identity N/A

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

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

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

      6. sub0-neg N/A

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

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

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

      8. *-lowering-*.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 98.5%

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

   9. Applied egg-rr 98.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 50000:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{elif}\;m \leq 1.36 \cdot 10^{+153}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(0 - m\right)}{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-146}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(0 - m\right)}{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.6e-146)
   (- 0.0 m)
   (if (<= m 1.0) (* m (/ m v)) (/ (* m (- 0.0 m)) m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-146) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (m * (0.0 - 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 <= 1.6d-146) then
        tmp = 0.0d0 - m
    else if (m <= 1.0d0) then
        tmp = m * (m / v)
    else
        tmp = (m * (0.0d0 - m)) / m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.6e-146) {
		tmp = 0.0 - m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = (m * (0.0 - m)) / m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.6e-146:
		tmp = 0.0 - m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = (m * (0.0 - m)) / m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.6e-146)
		tmp = Float64(0.0 - m);
	elseif (m <= 1.0)
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(Float64(m * Float64(0.0 - m)) / m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.6e-146)
		tmp = 0.0 - m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = (m * (0.0 - m)) / m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.6e-146], N[(0.0 - m), $MachinePrecision], If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(0.0 - m), $MachinePrecision]), $MachinePrecision] / m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 1.6e-146

   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 73.6%

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

   5. Simplified 73.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 73.6%

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

   7. Applied egg-rr 73.6%

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

   if 1.6e-146 < 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 96.7%

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

   5. Simplified 96.7%

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

   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 69.7%

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

   8. Simplified 69.7%

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

   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 5.8%

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

   5. Simplified 5.8%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. sub0-neg N/A

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

      4. +-lft-identity N/A

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

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

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

      6. sub0-neg N/A

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

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

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

      8. *-lowering-*.f64 54.9%

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

   7. Applied egg-rr 54.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 54.9%

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

   9. Applied egg-rr 54.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.6 \cdot 10^{-146}:\\ \;\;\;\;0 - m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(0 - m\right)}{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-8) (* m (+ (/ m v) -1.0)) (/ (* m (* m (- 1.0 m))) v)))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-8) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 3.7d-8) then
        tmp = m * ((m / v) + (-1.0d0))
    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 <= 3.7e-8) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = (m * (m * (1.0 - m))) / v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.7e-8:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = (m * (m * (1.0 - m))) / v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-8)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(m * Float64(m * Float64(1.0 - m))) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.7e-8)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = (m * (m * (1.0 - m))) / v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-8], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7e-8

   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 99.3%

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

   5. Simplified 99.3%

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

   if 3.7e-8 < 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

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

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

         \[\leadsto \frac{1}{m \cdot v} \cdot {m}^{3} + \left(\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{3}\right)\right) \]

      4. *-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

         \[\leadsto \left({m}^{2} \cdot m\right) \cdot \frac{1}{m \cdot v} + \left(\mathsf{neg}\left(\frac{\color{blue}{1}}{v} \cdot {m}^{3}\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto {m}^{2} \cdot \left(m \cdot \frac{1}{m \cdot v}\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{v} \cdot {m}^{3}}\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto {m}^{2} \cdot \left(m \cdot \frac{\frac{1}{m}}{v}\right) + \left(\mathsf{neg}\left(\frac{1}{v} \cdot {m}^{\color{blue}{3}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto {m}^{2} \cdot \frac{m \cdot \frac{1}{m}}{v} + \left(\mathsf{neg}\left(\frac{1}{v} \cdot \color{blue}{{m}^{3}}\right)\right) \]

      10. rgt-mult-inverse N/A

          \[\leadsto {m}^{2} \cdot \frac{1}{v} + \left(\mathsf{neg}\left(\frac{1}{v} \cdot {\color{blue}{m}}^{3}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{v} \cdot {m}^{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{v} \cdot {m}^{3}}\right)\right) \]

      12. cube-mult N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

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

      18. mul-1-neg N/A

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

   5. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.7e-8) (* m (+ (/ m v) -1.0)) (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.7e-8) {
		tmp = m * ((m / v) + -1.0);
	} 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 <= 3.7d-8) then
        tmp = m * ((m / v) + (-1.0d0))
    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 <= 3.7e-8) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3.7e-8:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = m * ((m * (1.0 - m)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3.7e-8)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.7e-8)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = m * ((m * (1.0 - m)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3.7e-8], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7e-8

   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 99.3%

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

   5. Simplified 99.3%

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

   if 3.7e-8 < 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%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   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 98.5%

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

   5. Simplified 98.5%

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

   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 5.8%

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

   5. Simplified 5.8%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. sub0-neg N/A

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

      4. +-lft-identity N/A

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

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

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

      6. sub0-neg N/A

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

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

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

      8. *-lowering-*.f64 54.9%

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

   7. Applied egg-rr 54.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 54.9%

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

   9. Applied egg-rr 54.9%

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

4. Final simplification 77.9%

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

Alternative 7: 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 8: 38.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 1.15 \cdot 10^{-148}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= v 1.15e-148) (* m (/ m v)) (- 0.0 m)))

C

double code(double m, double v) {
	double tmp;
	if (v <= 1.15e-148) {
		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 <= 1.15d-148) 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 <= 1.15e-148) {
		tmp = m * (m / v);
	} else {
		tmp = 0.0 - m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if v <= 1.15e-148:
		tmp = m * (m / v)
	else:
		tmp = 0.0 - m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (v <= 1.15e-148)
		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 <= 1.15e-148)
		tmp = m * (m / v);
	else
		tmp = 0.0 - m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[v, 1.15e-148], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], N[(0.0 - m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 1.14999999999999999e-148

   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 53.4%

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

   5. Simplified 53.4%

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

   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 37.8%

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

   8. Simplified 37.8%

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

   if 1.14999999999999999e-148 < 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 46.2%

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

   5. Simplified 46.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 46.2%

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

   7. Applied egg-rr 46.2%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 1.15 \cdot 10^{-148}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;0 - m\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.4% 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 31.0%

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

5. Simplified 31.0%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 31.0%

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

7. Applied egg-rr 31.0%

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

8. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2024138 
(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))