a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 13

Speedup: 1.0×

• 42.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 10^{-9}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v + m \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1e-9)
   (* m (+ -1.0 (/ m (+ v (* m v)))))
   (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1e-9) {
		tmp = m * (-1.0 + (m / (v + (m * v))));
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1d-9) then
        tmp = m * ((-1.0d0) + (m / (v + (m * v))))
    else
        tmp = m * ((m * (1.0d0 - m)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1e-9) {
		tmp = m * (-1.0 + (m / (v + (m * v))));
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1e-9:
		tmp = m * (-1.0 + (m / (v + (m * v))))
	else:
		tmp = m * ((m * (1.0 - m)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1e-9)
		tmp = Float64(m * Float64(-1.0 + Float64(m / Float64(v + Float64(m * v)))));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1e-9)
		tmp = m * (-1.0 + (m / (v + (m * v))));
	else
		tmp = m * ((m * (1.0 - m)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1e-9], N[(m * N[(-1.0 + N[(m / N[(v + N[(m * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 1.00000000000000006e-9

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 85.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-/l* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 99.9%

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

   if 1.00000000000000006e-9 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in v around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 10^{-9}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v + m \cdot v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 2: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-181}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 7.4 \cdot 10^{-157}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 2.5 \cdot 10^{-146}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2e-181)
   (- m)
   (if (<= m 7.4e-157)
     (* m (/ m v))
     (if (<= m 2.5e-146)
       (- m)
       (if (<= m 1.0) (/ m (/ v m)) (* m (/ m (- v))))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-181) {
		tmp = -m;
	} else if (m <= 7.4e-157) {
		tmp = m * (m / v);
	} else if (m <= 2.5e-146) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (m / -v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.2d-181) then
        tmp = -m
    else if (m <= 7.4d-157) then
        tmp = m * (m / v)
    else if (m <= 2.5d-146) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = m * (m / -v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.2e-181) {
		tmp = -m;
	} else if (m <= 7.4e-157) {
		tmp = m * (m / v);
	} else if (m <= 2.5e-146) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = m * (m / -v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.2e-181:
		tmp = -m
	elif m <= 7.4e-157:
		tmp = m * (m / v)
	elif m <= 2.5e-146:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = m * (m / -v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.2e-181)
		tmp = Float64(-m);
	elseif (m <= 7.4e-157)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 2.5e-146)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(m / Float64(-v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.2e-181)
		tmp = -m;
	elseif (m <= 7.4e-157)
		tmp = m * (m / v);
	elseif (m <= 2.5e-146)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = m * (m / -v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.2e-181], (-m), If[LessEqual[m, 7.4e-157], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.5e-146], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / (-v)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 2.19999999999999997e-181 or 7.3999999999999995e-157 < m < 2.49999999999999979e-146

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 83.3%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 83.3%

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

      1. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

   if 2.19999999999999997e-181 < m < 7.3999999999999995e-157

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 33.8%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 18.5%

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

      1. associate-/l* 18.5%

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

      2. unpow2 18.5%

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

      3. associate-*r/ 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in m around 0 81.5%

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

   if 2.49999999999999979e-146 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 99.6%

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

      6. associate-*r/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 80.7%

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

      1. associate-/l* 80.8%

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

      2. unpow2 80.8%

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

      3. associate-*r/ 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in m around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/r/ 78.4%

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

   9. Applied egg-rr 78.4%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 78.8%

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

      6. frac-2neg 78.8%

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

      7. associate-*r/ 78.8%

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

      8. sqr-neg 78.8%

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

   9. Applied egg-rr 78.8%

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

       1. associate-/l* 78.8%

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

       2. associate-/r/ 78.8%

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

   11. Simplified 78.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-181}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 7.4 \cdot 10^{-157}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 2.5 \cdot 10^{-146}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{-v}\\ \end{array} \]

Alternative 3: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-184}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 10^{-156}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 7 \cdot 10^{-149}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{-v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 7.2e-184)
   (- m)
   (if (<= m 1e-156)
     (* m (/ m v))
     (if (<= m 7e-149)
       (- m)
       (if (<= m 1.0) (/ m (/ v m)) (/ (* m m) (- v)))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-184) {
		tmp = -m;
	} else if (m <= 1e-156) {
		tmp = m * (m / v);
	} else if (m <= 7e-149) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m * m) / -v;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 7.2d-184) then
        tmp = -m
    else if (m <= 1d-156) then
        tmp = m * (m / v)
    else if (m <= 7d-149) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = (m * m) / -v
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 7.2e-184) {
		tmp = -m;
	} else if (m <= 1e-156) {
		tmp = m * (m / v);
	} else if (m <= 7e-149) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = (m * m) / -v;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 7.2e-184:
		tmp = -m
	elif m <= 1e-156:
		tmp = m * (m / v)
	elif m <= 7e-149:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = (m * m) / -v
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 7.2e-184)
		tmp = Float64(-m);
	elseif (m <= 1e-156)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 7e-149)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(Float64(m * m) / Float64(-v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 7.2e-184)
		tmp = -m;
	elseif (m <= 1e-156)
		tmp = m * (m / v);
	elseif (m <= 7e-149)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = (m * m) / -v;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 7.2e-184], (-m), If[LessEqual[m, 1e-156], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7e-149], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], N[(N[(m * m), $MachinePrecision] / (-v)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if m < 7.2000000000000002e-184 or 1.00000000000000004e-156 < m < 7e-149

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 83.3%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 83.3%

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

      1. neg-mul-1 83.3%

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

   6. Simplified 83.3%

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

   if 7.2000000000000002e-184 < m < 1.00000000000000004e-156

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 33.8%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 18.5%

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

      1. associate-/l* 18.5%

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

      2. unpow2 18.5%

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

      3. associate-*r/ 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in m around 0 81.5%

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

   if 7e-149 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 99.6%

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

      6. associate-*r/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 80.7%

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

      1. associate-/l* 80.8%

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

      2. unpow2 80.8%

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

      3. associate-*r/ 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in m around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/r/ 78.4%

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

   9. Applied egg-rr 78.4%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 78.8%

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

      6. frac-2neg 78.8%

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

      7. associate-*r/ 78.8%

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

      8. sqr-neg 78.8%

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

   9. Applied egg-rr 78.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 7.2 \cdot 10^{-184}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 10^{-156}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 7 \cdot 10^{-149}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{-v}\\ \end{array} \]

Alternative 4: 37.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-184}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-156} \lor \neg \left(m \leq 8.6 \cdot 10^{-147}\right) \land m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3e-184)
   (- m)
   (if (or (<= m 1.05e-156) (and (not (<= m 8.6e-147)) (<= m 1.0)))
     (* m (/ m v))
     (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3e-184) {
		tmp = -m;
	} else if ((m <= 1.05e-156) || (!(m <= 8.6e-147) && (m <= 1.0))) {
		tmp = m * (m / v);
	} else {
		tmp = -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 <= 3d-184) then
        tmp = -m
    else if ((m <= 1.05d-156) .or. (.not. (m <= 8.6d-147)) .and. (m <= 1.0d0)) then
        tmp = m * (m / v)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 3e-184) {
		tmp = -m;
	} else if ((m <= 1.05e-156) || (!(m <= 8.6e-147) && (m <= 1.0))) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 3e-184:
		tmp = -m
	elif (m <= 1.05e-156) or (not (m <= 8.6e-147) and (m <= 1.0)):
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 3e-184)
		tmp = Float64(-m);
	elseif ((m <= 1.05e-156) || (!(m <= 8.6e-147) && (m <= 1.0)))
		tmp = Float64(m * Float64(m / v));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3e-184)
		tmp = -m;
	elseif ((m <= 1.05e-156) || (~((m <= 8.6e-147)) && (m <= 1.0)))
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 3e-184], (-m), If[Or[LessEqual[m, 1.05e-156], And[N[Not[LessEqual[m, 8.6e-147]], $MachinePrecision], LessEqual[m, 1.0]]], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 2.99999999999999991e-184 or 1.05000000000000006e-156 < m < 8.6000000000000002e-147 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 94.8%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 29.5%

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

      1. neg-mul-1 29.5%

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

   6. Simplified 29.5%

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

   if 2.99999999999999991e-184 < m < 1.05000000000000006e-156 or 8.6000000000000002e-147 < m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 88.3%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-/l* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 70.1%

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

      1. associate-/l* 70.1%

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

      2. unpow2 70.1%

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

      3. associate-*r/ 80.9%

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

   6. Simplified 80.9%

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

   7. Taylor expanded in m around 0 78.9%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-184}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-156} \lor \neg \left(m \leq 8.6 \cdot 10^{-147}\right) \land m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 5: 37.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.45 \cdot 10^{-156}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-181)
   (- m)
   (if (<= m 1.45e-156)
     (* m (/ m v))
     (if (<= m 1.5e-146) (- m) (if (<= m 1.0) (/ m (/ v m)) (- m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-181) {
		tmp = -m;
	} else if (m <= 1.45e-156) {
		tmp = m * (m / v);
	} else if (m <= 1.5e-146) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.1d-181) then
        tmp = -m
    else if (m <= 1.45d-156) then
        tmp = m * (m / v)
    else if (m <= 1.5d-146) then
        tmp = -m
    else if (m <= 1.0d0) then
        tmp = m / (v / m)
    else
        tmp = -m
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-181) {
		tmp = -m;
	} else if (m <= 1.45e-156) {
		tmp = m * (m / v);
	} else if (m <= 1.5e-146) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m / (v / m);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-181:
		tmp = -m
	elif m <= 1.45e-156:
		tmp = m * (m / v)
	elif m <= 1.5e-146:
		tmp = -m
	elif m <= 1.0:
		tmp = m / (v / m)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-181)
		tmp = Float64(-m);
	elseif (m <= 1.45e-156)
		tmp = Float64(m * Float64(m / v));
	elseif (m <= 1.5e-146)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(-m);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-181)
		tmp = -m;
	elseif (m <= 1.45e-156)
		tmp = m * (m / v);
	elseif (m <= 1.5e-146)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m / (v / m);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-181], (-m), If[LessEqual[m, 1.45e-156], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.5e-146], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]]]]

Derivation

1. Split input into 3 regimes

2. if m < 2.10000000000000003e-181 or 1.4500000000000001e-156 < m < 1.50000000000000009e-146 or 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 94.8%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 29.5%

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

      1. neg-mul-1 29.5%

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

   6. Simplified 29.5%

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

   if 2.10000000000000003e-181 < m < 1.4500000000000001e-156

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 33.8%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 18.5%

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

      1. associate-/l* 18.5%

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

      2. unpow2 18.5%

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

      3. associate-*r/ 81.5%

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

   6. Simplified 81.5%

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

   7. Taylor expanded in m around 0 81.5%

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

   if 1.50000000000000009e-146 < m < 1

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 99.6%

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

      6. associate-*r/ 99.6%

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

      7. *-lft-identity 99.6%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.6%

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

      12. associate-*r/ 99.7%

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

      13. associate-/l* 99.7%

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

      14. /-rgt-identity 99.7%

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

      15. associate-/l* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in v around 0 80.7%

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

      1. associate-/l* 80.8%

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

      2. unpow2 80.8%

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

      3. associate-*r/ 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in m around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/r/ 78.4%

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

   9. Applied egg-rr 78.4%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-181}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.45 \cdot 10^{-156}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 1.5 \cdot 10^{-146}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 2.1e-14) (* m (+ -1.0 (/ m v))) (* m (/ m (/ v (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 2.1d-14) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * (m / (v / (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.1e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m / (v / (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 2.1e-14:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (m / (v / (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 2.1e-14)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(m / Float64(v / Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 2.1e-14)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * (m / (v / (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 2.1e-14], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.0999999999999999e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 99.8%

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

   if 2.0999999999999999e-14 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.8%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 7: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.95e-14) (* m (+ -1.0 (/ m v))) (* m (/ (* m (- 1.0 m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.95e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.95d-14) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * ((m * (1.0d0 - m)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.95e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * ((m * (1.0 - m)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 1.95e-14:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * ((m * (1.0 - m)) / v)
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 1.95e-14)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.95e-14)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * ((m * (1.0 - m)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 1.95e-14], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $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 < 1.9499999999999999e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 99.8%

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

   if 1.9499999999999999e-14 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.8%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-out 99.9%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in v around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l/ 93.1%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.8%

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

   9. associate-*r* 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-out 99.8%

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

   12. associate-*r/ 99.9%

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

   13. associate-/l* 99.9%

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

   14. /-rgt-identity 99.9%

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

   15. associate-/l* 99.9%

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 9: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 86.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. unpow2 84.9%

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

      3. associate-*r/ 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. associate-*r/ 84.9%

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

      7. associate-/l* 98.7%

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

   8. Simplified 98.7%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unpow2 98.1%

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

      4. distribute-rgt-neg-out 98.1%

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

      5. associate-*l/ 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 98.4%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 86.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. unpow2 84.9%

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

      3. associate-*r/ 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. associate-*r/ 84.9%

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

      7. associate-/l* 98.7%

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

   8. Simplified 98.7%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. neg-mul-1 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 98.4%

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

Alternative 11: 87.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (m * 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(Float64(m * m) / Float64(-v));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

   2. Taylor expanded in m around 0 98.7%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 78.8%

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

      6. frac-2neg 78.8%

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

      7. associate-*r/ 78.8%

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

      8. sqr-neg 78.8%

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

   9. Applied egg-rr 78.8%

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

4. Final simplification 88.7%

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

Alternative 12: 87.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	tmp = 0
	if m <= 1.0:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * 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(Float64(m * m) / Float64(-v));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l/ 86.1%

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

      6. associate-*r/ 99.8%

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

      7. *-lft-identity 99.8%

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

      8. associate-*l/ 99.7%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-/l* 99.9%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. neg-mul-1 99.8%

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

      4. associate-/l* 99.8%

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

      5. div-inv 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*l* 99.7%

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

      8. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in m around 0 84.9%

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

      1. +-commutative 84.9%

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

      2. unpow2 84.9%

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

      3. associate-*r/ 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. associate-*r/ 84.9%

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

      7. associate-/l* 98.7%

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

   8. Simplified 98.7%

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

   if 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      7. *-lft-identity 99.9%

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

      8. associate-*l/ 99.9%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. distribute-rgt-out 100.0%

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

      12. associate-*r/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-/l* 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

      3. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. add-sqr-sqrt 0.1%

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

      2. sqrt-unprod 0.1%

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

      3. sqr-neg 0.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 78.8%

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

      6. frac-2neg 78.8%

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

      7. associate-*r/ 78.8%

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

      8. sqr-neg 78.8%

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

   9. Applied egg-rr 78.8%

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

4. Final simplification 88.7%

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

Alternative 13: 27.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*l/ 93.1%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.8%

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

   9. associate-*r* 99.8%

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

   10. *-commutative 99.8%

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

   11. distribute-rgt-out 99.8%

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

   12. associate-*r/ 99.9%

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

   13. associate-/l* 99.9%

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

   14. /-rgt-identity 99.9%

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

   15. associate-/l* 99.9%

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 25.7%

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

   1. neg-mul-1 25.7%

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

6. Simplified 25.7%

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

7. Final simplification 25.7%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023256 
(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))