a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 12

Speedup: 1.0×

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

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. 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/ 95.1%

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

   6. associate-*r/ 99.9%

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

   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/ 99.8%

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

   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/ 99.8%

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

   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%

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 8.5e-14) (- (/ m (/ v m)) m) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 8.5d-14) then
        tmp = (m / (v / m)) - m
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 8.5e-14) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 8.5e-14:
		tmp = (m / (v / m)) - m
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 8.5e-14)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(m * Float64(m * Float64(Float64(1.0 - m) / v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 8.5e-14)
		tmp = (m / (v / m)) - m;
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 8.5e-14], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8.50000000000000038e-14

   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 99.8%

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

      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/ 90.2%

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

      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/ 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.8%

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

      2. *-commutative 99.8%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

      5. *-commutative 99.7%

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

      6. neg-mul-1 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 99.6%

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

      1. associate-*r/ 90.1%

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

      2. associate-/l* 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 8.50000000000000038e-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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r/ 99.9%

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

      4. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5e-163) (- m) (if (<= m 1.0) (/ m (/ v m)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5e-163) {
		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 <= 5d-163) 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 <= 5e-163) {
		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 <= 5e-163:
		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 <= 5e-163)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(m * Float64(Float64(-m) / v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5e-163)
		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, 5e-163], (-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 3 regimes

2. if m < 4.99999999999999977e-163

   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/ 82.3%

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

      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 99.9%

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

      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%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 82.3%

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

      1. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

   if 4.99999999999999977e-163 < m < 1

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l/ 97.5%

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

      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/ 99.5%

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

      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/ 99.6%

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

      13. associate-/l* 99.6%

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

      14. /-rgt-identity 99.6%

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

      15. associate-*l/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r* 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*l/ 83.0%

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

      5. associate-*r/ 83.0%

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

      6. *-commutative 83.0%

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

      7. associate-*l* 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in m around 0 77.7%

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

      1. unpow2 77.7%

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

      2. associate-/l* 80.0%

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

   9. Simplified 80.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. distribute-frac-neg 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 80.2%

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

      8. distribute-lft-neg-in 80.2%

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

      9. frac-2neg 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. associate-*r/ 80.2%

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

       2. distribute-lft-neg-in 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 4: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 3.6e-163) (- m) (if (<= m 1.0) (/ m (/ v m)) (/ (* m (- m)) v))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 3.6e-163) {
		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 <= 3.6d-163) 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 <= 3.6e-163) {
		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 <= 3.6e-163:
		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 <= 3.6e-163)
		tmp = Float64(-m);
	elseif (m <= 1.0)
		tmp = Float64(m / Float64(v / m));
	else
		tmp = Float64(Float64(m * Float64(-m)) / v);
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.6e-163)
		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, 3.6e-163], (-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 3 regimes

2. if m < 3.5999999999999998e-163

   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/ 82.3%

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

      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 99.9%

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

      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%

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

      16. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 82.3%

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

      1. neg-mul-1 82.3%

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

   6. Simplified 82.3%

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

   if 3.5999999999999998e-163 < m < 1

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l/ 97.5%

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

      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/ 99.5%

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

      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/ 99.6%

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

      13. associate-/l* 99.6%

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

      14. /-rgt-identity 99.6%

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

      15. associate-*l/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r* 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*l/ 83.0%

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

      5. associate-*r/ 83.0%

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

      6. *-commutative 83.0%

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

      7. associate-*l* 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in m around 0 77.7%

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

      1. unpow2 77.7%

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

      2. associate-/l* 80.0%

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

   9. Simplified 80.0%

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

   if 1 < m

   1. Initial program 99.9%

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   2. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. distribute-frac-neg 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 80.2%

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

      8. distribute-lft-neg-in 80.2%

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

      9. frac-2neg 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. distribute-neg-frac 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m\right)}{v}\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.0)
		tmp = (m * (m / v)) - 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[(m / v), $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/ 90.6%

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

      6. associate-*r/ 99.7%

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

      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/ 99.7%

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

      9. associate-*r* 99.8%

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

      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 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-commutative 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

      5. *-commutative 99.7%

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

      6. neg-mul-1 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around 0 90.6%

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

      1. mul-1-neg 90.6%

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

      2. +-commutative 90.6%

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

      3. unsub-neg 90.6%

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

      4. unpow2 90.6%

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

      5. associate-/l* 90.5%

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

      6. associate-/r/ 90.5%

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

   8. Simplified 90.5%

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

   9. Taylor expanded in m around 0 88.7%

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

       1. +-commutative 88.7%

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

       2. unpow2 88.7%

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

       3. mul-1-neg 88.7%

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

       4. sub-neg 88.7%

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

       5. associate-*r/ 97.9%

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

   11. Simplified 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

   9. Simplified 97.9%

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

       1. frac-2neg 97.9%

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

       2. remove-double-neg 97.9%

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

       3. associate-/r/ 97.9%

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

       4. associate-/l* 97.9%

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

   11. Applied egg-rr 97.9%

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

4. Final simplification 97.9%

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

Alternative 6: 98.2% 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(Float64(-m) / 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/ 90.6%

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

      6. associate-*r/ 99.7%

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

      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/ 99.7%

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

      9. associate-*r* 99.8%

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

      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 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-commutative 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

      5. *-commutative 99.7%

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

      6. neg-mul-1 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 97.9%

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

      1. associate-*r/ 88.7%

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

      2. associate-/l* 97.9%

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

   8. Applied egg-rr 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

   9. Simplified 97.9%

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

       1. frac-2neg 97.9%

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

       2. remove-double-neg 97.9%

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

       3. associate-/r/ 97.9%

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

       4. associate-/l* 97.9%

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

   11. Applied egg-rr 97.9%

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

4. Final simplification 97.9%

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

Alternative 7: 98.2% accurate, 1.1× speedup

Math

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

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(Float64(m * 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[(N[(m * m), $MachinePrecision] / N[((-v) / m), $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/ 90.6%

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

      6. associate-*r/ 99.7%

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

      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/ 99.7%

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

      9. associate-*r* 99.8%

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

      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 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-commutative 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

      5. *-commutative 99.7%

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

      6. neg-mul-1 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in m around 0 97.9%

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

      1. associate-*r/ 88.7%

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

      2. associate-/l* 97.9%

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

   8. Applied egg-rr 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. neg-mul-1 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 8: 38.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{-165}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.6e-165) (- m) (if (<= m 1.0) (* m (/ m v)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.6e-165) {
		tmp = -m;
	} else if (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 <= 5.6d-165) then
        tmp = -m
    else if (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 <= 5.6e-165) {
		tmp = -m;
	} else if (m <= 1.0) {
		tmp = m * (m / v);
	} else {
		tmp = -m;
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 5.6e-165:
		tmp = -m
	elif m <= 1.0:
		tmp = m * (m / v)
	else:
		tmp = -m
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 5.6e-165)
		tmp = Float64(-m);
	elseif (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 <= 5.6e-165)
		tmp = -m;
	elseif (m <= 1.0)
		tmp = m * (m / v);
	else
		tmp = -m;
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 5.6e-165], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5999999999999999e-165 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 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/ 94.2%

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

      6. associate-*r/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 30.6%

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

      1. neg-mul-1 30.6%

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

   6. Simplified 30.6%

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

   if 5.5999999999999999e-165 < m < 1

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l/ 97.5%

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

      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/ 99.5%

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

      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/ 99.6%

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

      13. associate-/l* 99.6%

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

      14. /-rgt-identity 99.6%

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

      15. associate-*l/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. unpow2 80.9%

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

      2. sqr-neg 80.9%

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

      3. associate-/l* 80.8%

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

      4. sqr-neg 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in m around 0 77.7%

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

      1. associate-*l/ 79.9%

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

   9. Applied egg-rr 79.9%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{-165}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9: 38.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.8e-163) (- m) (if (<= m 1.0) (/ m (/ v m)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.8e-163) {
		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 <= 4.8d-163) 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 <= 4.8e-163) {
		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 <= 4.8e-163:
		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 <= 4.8e-163)
		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 <= 4.8e-163)
		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, 4.8e-163], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 4.8000000000000001e-163 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 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/ 94.2%

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

      6. associate-*r/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 30.6%

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

      1. neg-mul-1 30.6%

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

   6. Simplified 30.6%

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

   if 4.8000000000000001e-163 < m < 1

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l/ 97.5%

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

      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/ 99.5%

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

      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/ 99.6%

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

      13. associate-/l* 99.6%

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

      14. /-rgt-identity 99.6%

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

      15. associate-*l/ 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in v around 0 80.9%

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

      1. unpow2 80.9%

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

      2. associate-*r* 80.9%

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

      3. *-commutative 80.9%

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

      4. associate-*l/ 83.0%

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

      5. associate-*r/ 83.0%

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

      6. *-commutative 83.0%

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

      7. associate-*l* 80.7%

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

   6. Simplified 80.7%

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

   7. Taylor expanded in m around 0 77.7%

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

      1. unpow2 77.7%

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

      2. associate-/l* 80.0%

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

   9. Simplified 80.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10: 88.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(m * (-m)), $MachinePrecision] / 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/ 90.6%

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

      6. associate-*r/ 99.7%

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

      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/ 99.7%

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

      9. associate-*r* 99.8%

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

      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 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in m around 0 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. distribute-frac-neg 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 80.2%

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

      8. distribute-lft-neg-in 80.2%

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

      9. frac-2neg 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. distribute-neg-frac 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 89.3%

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

Alternative 11: 88.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(N[(m * N[(m / v), $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/ 90.6%

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

      6. associate-*r/ 99.7%

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

      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/ 99.7%

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

      9. associate-*r* 99.8%

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

      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 99.7%

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

      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/ 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-commutative 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.7%

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

      5. *-commutative 99.7%

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

      6. neg-mul-1 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in v around 0 90.6%

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

      1. mul-1-neg 90.6%

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

      2. +-commutative 90.6%

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

      3. unsub-neg 90.6%

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

      4. unpow2 90.6%

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

      5. associate-/l* 90.5%

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

      6. associate-/r/ 90.5%

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

   8. Simplified 90.5%

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

   9. Taylor expanded in m around 0 88.7%

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

       1. +-commutative 88.7%

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

       2. unpow2 88.7%

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

       3. mul-1-neg 88.7%

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

       4. sub-neg 88.7%

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

       5. associate-*r/ 97.9%

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

   11. Simplified 97.9%

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

   if 1 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. 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/ 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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/ 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   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. unpow2 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. sqr-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. frac-2neg 0.1%

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

      2. distribute-frac-neg 0.1%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-prod 0.1%

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

      5. sqr-neg 0.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 80.2%

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

      8. distribute-lft-neg-in 80.2%

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

      9. frac-2neg 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. distribute-neg-frac 80.2%

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

   11. Simplified 80.2%

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

4. Final simplification 89.3%

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

Alternative 12: 27.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(m, v):
	return -m

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. sub-neg 99.8%

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

   3. 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/ 95.1%

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

   6. associate-*r/ 99.9%

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

   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/ 99.8%

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

   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/ 99.8%

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

   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%

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

   16. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in m around 0 26.6%

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

   1. neg-mul-1 26.6%

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

6. Simplified 26.6%

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

7. Final simplification 26.6%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023175 
(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))