a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 10

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
2. 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/ 93.5%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.7%

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

   9. associate-*r* 99.7%

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

   10. *-commutative 99.7%

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

   11. distribute-rgt-out 99.7%

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

   12. associate-*r/ 99.8%

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

   13. associate-/l* 99.8%

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

   14. /-rgt-identity 99.8%

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

   15. associate-/l* 99.8%

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4e-17) (* m (+ -1.0 (/ m v))) (* m (* m (/ (- 1.0 m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4d-17) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m * (m * ((1.0d0 - m) / v))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m * (m * ((1.0 - m) / v));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4e-17:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m * (m * ((1.0 - m) / v))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4e-17)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	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 <= 4e-17)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m * (m * ((1.0 - m) / v));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4e-17], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.00000000000000029e-17

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 88.2%

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

      6. associate-*r/ 99.7%

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

      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* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 99.8%

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

   if 4.00000000000000029e-17 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      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/ 99.9%

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

      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 v around 0 99.4%

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

      1. associate-/l* 99.3%

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

      2. unpow2 99.3%

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

   6. Simplified 99.3%

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

      1. associate-/l* 99.4%

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

      2. div-inv 99.3%

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

      3. clear-num 99.4%

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

      4. div-inv 99.3%

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

      5. clear-num 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.1e-17) (* m (+ -1.0 (/ m v))) (/ m (/ v (* m (- 1.0 m))))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.1e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 4.1d-17) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = m / (v / (m * (1.0d0 - m)))
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	double tmp;
	if (m <= 4.1e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = m / (v / (m * (1.0 - m)));
	}
	return tmp;
}

Python

def code(m, v):
	tmp = 0
	if m <= 4.1e-17:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = m / (v / (m * (1.0 - m)))
	return tmp

Julia

function code(m, v)
	tmp = 0.0
	if (m <= 4.1e-17)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(m / Float64(v / Float64(m * Float64(1.0 - m))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 4.1e-17)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = m / (v / (m * (1.0 - m)));
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := If[LessEqual[m, 4.1e-17], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m / N[(v / N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.1000000000000001e-17

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 88.2%

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

      6. associate-*r/ 99.7%

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

      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* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-/l* 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. associate-*l/ 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in m around 0 99.8%

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

   if 4.1000000000000001e-17 < m

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 99.9%

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

      6. associate-*r/ 99.9%

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

      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/ 99.9%

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

      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. Step-by-step derivation

      1. flip3-+ 3.0%

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

      2. associate-*r/ 2.2%

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

      3. pow3 2.1%

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

      4. metadata-eval 2.1%

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

      5. +-commutative 2.1%

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

      6. pow3 2.2%

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

      7. associate-/r/ 2.1%

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

      8. div-inv 2.1%

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

      9. clear-num 2.2%

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

   5. Applied egg-rr 2.2%

      \[\leadsto \color{blue}{\frac{m \cdot \left(-1 + {\left(m \cdot \frac{1 - m}{v}\right)}^{3}\right)}{{\left(m \cdot \frac{1 - m}{v}\right)}^{2} + \left(1 - \frac{-m}{\frac{v}{1 - m}}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 3.0%

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

      2. associate-*r/ 3.0%

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

      3. associate-/l* 2.9%

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

      4. associate-/r/ 2.9%

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

      5. distribute-neg-frac 2.9%

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

      6. cancel-sign-sub 2.9%

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

      7. associate-/r/ 2.9%

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

      8. associate-*r/ 3.0%

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

      9. associate-/l* 3.0%

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

   7. Simplified 3.0%

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

   8. Taylor expanded in v around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := N[(m * N[(-1.0 + N[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]), $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/ 93.5%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.7%

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

   9. associate-*r* 99.7%

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

   10. *-commutative 99.7%

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

   11. distribute-rgt-out 99.7%

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

   12. associate-*r/ 99.8%

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

   13. associate-/l* 99.8%

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

   14. /-rgt-identity 99.8%

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

   15. associate-*l/ 99.8%

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 5: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 4.5e-179) (- m) (if (<= m 1.0) (/ m (/ v m)) (* m (/ (- m) v)))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 4.5e-179) {
		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 <= 4.5d-179) 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 <= 4.5e-179) {
		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 <= 4.5e-179:
		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 <= 4.5e-179)
		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 <= 4.5e-179)
		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, 4.5e-179], (-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 < 4.49999999999999992e-179

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l/ 81.2%

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

      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/ 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/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 100.0%

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

      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 81.2%

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

      1. neg-mul-1 81.2%

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

   6. Simplified 81.2%

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

   if 4.49999999999999992e-179 < 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/ 93.7%

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

      6. associate-*r/ 99.5%

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

      7. *-lft-identity 99.5%

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

      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* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out 99.5%

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

      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/ 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. flip3-+ 65.4%

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

      2. associate-*r/ 65.4%

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

      3. pow3 65.3%

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

      4. metadata-eval 65.3%

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

      5. +-commutative 65.3%

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

      6. pow3 65.4%

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

      7. associate-/r/ 65.3%

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

      8. div-inv 65.1%

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

      9. clear-num 65.2%

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

   5. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{\frac{m \cdot \left(-1 + {\left(m \cdot \frac{1 - m}{v}\right)}^{3}\right)}{{\left(m \cdot \frac{1 - m}{v}\right)}^{2} + \left(1 - \frac{-m}{\frac{v}{1 - m}}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 65.2%

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

      2. associate-*r/ 65.1%

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

      3. associate-/l* 65.1%

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

      4. associate-/r/ 65.1%

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

      5. distribute-neg-frac 65.1%

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

      6. cancel-sign-sub 65.1%

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

      7. associate-/r/ 65.1%

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

      8. associate-*r/ 65.2%

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

      9. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in v around 0 72.4%

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

   9. Taylor expanded in m around 0 69.6%

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

   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/ 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* 99.8%

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

      10. *-commutative 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      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 v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

   9. Simplified 0.1%

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

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-unprod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 79.0%

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

       6. neg-mul-1 79.0%

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

       7. *-un-lft-identity 79.0%

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

       8. times-frac 79.0%

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

       9. metadata-eval 79.0%

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

   11. Applied egg-rr 79.0%

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

       1. mul-1-neg 79.0%

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

       2. associate-*r/ 79.0%

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

       3. distribute-rgt-neg-in 79.0%

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

       4. distribute-frac-neg 79.0%

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

   13. Simplified 79.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{v}\\ \end{array} \]

Alternative 6: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 1

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-lft-in 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 88.8%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.7%

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

      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 97.6%

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

   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/ 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* 99.8%

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

      10. *-commutative 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      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 v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. associate-/l* 99.9%

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

      2. div-inv 99.9%

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

      3. clear-num 99.9%

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

      4. div-inv 99.9%

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

      5. clear-num 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in m around inf 98.3%

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

       1. mul-1-neg 98.3%

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

       2. unpow2 98.3%

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

       3. *-rgt-identity 98.3%

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

       4. associate-*r/ 98.2%

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

       5. associate-*l* 98.3%

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

       6. distribute-rgt-neg-in 98.3%

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

       7. associate-*r/ 98.3%

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

       8. *-rgt-identity 98.3%

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

       9. distribute-frac-neg 98.3%

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

       10. *-commutative 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 97.9%

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

Alternative 7: 38.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 5.9e-179) (- m) (if (<= m 1.0) (* m (/ m v)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 5.9e-179) {
		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 <= 5.9d-179) 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 <= 5.9e-179) {
		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 <= 5.9e-179:
		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 <= 5.9e-179)
		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 <= 5.9e-179)
		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, 5.9e-179], (-m), If[LessEqual[m, 1.0], N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 5.90000000000000029e-179 or 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 93.4%

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

      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* 99.8%

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

      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/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 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 31.7%

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

      1. neg-mul-1 31.7%

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

   6. Simplified 31.7%

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

   if 5.90000000000000029e-179 < 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/ 93.7%

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

      6. associate-*r/ 99.5%

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

      7. *-lft-identity 99.5%

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

      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* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out 99.5%

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

      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/ 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. flip3-+ 65.4%

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

      2. associate-*r/ 65.4%

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

      3. pow3 65.3%

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

      4. metadata-eval 65.3%

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

      5. +-commutative 65.3%

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

      6. pow3 65.4%

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

      7. associate-/r/ 65.3%

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

      8. div-inv 65.1%

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

      9. clear-num 65.2%

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

   5. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{\frac{m \cdot \left(-1 + {\left(m \cdot \frac{1 - m}{v}\right)}^{3}\right)}{{\left(m \cdot \frac{1 - m}{v}\right)}^{2} + \left(1 - \frac{-m}{\frac{v}{1 - m}}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 65.2%

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

      2. associate-*r/ 65.1%

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

      3. associate-/l* 65.1%

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

      4. associate-/r/ 65.1%

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

      5. distribute-neg-frac 65.1%

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

      6. cancel-sign-sub 65.1%

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

      7. associate-/r/ 65.1%

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

      8. associate-*r/ 65.2%

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

      9. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in v around 0 72.4%

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

   9. Taylor expanded in m around 0 63.9%

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

       1. unpow2 63.9%

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

       2. associate-*r/ 69.5%

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

   11. Simplified 69.5%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8: 38.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \end{array} \]

FPCore

(FPCore (m v)
 :precision binary64
 (if (<= m 1.7e-179) (- m) (if (<= m 1.0) (/ m (/ v m)) (- m))))

C

double code(double m, double v) {
	double tmp;
	if (m <= 1.7e-179) {
		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 <= 1.7d-179) 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 <= 1.7e-179) {
		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 <= 1.7e-179:
		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 <= 1.7e-179)
		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 <= 1.7e-179)
		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, 1.7e-179], (-m), If[LessEqual[m, 1.0], N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision], (-m)]]

Derivation

1. Split input into 2 regimes

2. if m < 1.6999999999999999e-179 or 1 < m

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l/ 93.4%

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

      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* 99.8%

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

      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/ 100.0%

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

      13. associate-/l* 100.0%

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

      14. /-rgt-identity 100.0%

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

      15. associate-*l/ 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 31.7%

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

      1. neg-mul-1 31.7%

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

   6. Simplified 31.7%

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

   if 1.6999999999999999e-179 < 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/ 93.7%

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

      6. associate-*r/ 99.5%

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

      7. *-lft-identity 99.5%

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

      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* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out 99.5%

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

      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/ 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. flip3-+ 65.4%

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

      2. associate-*r/ 65.4%

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

      3. pow3 65.3%

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

      4. metadata-eval 65.3%

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

      5. +-commutative 65.3%

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

      6. pow3 65.4%

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

      7. associate-/r/ 65.3%

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

      8. div-inv 65.1%

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

      9. clear-num 65.2%

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

   5. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{\frac{m \cdot \left(-1 + {\left(m \cdot \frac{1 - m}{v}\right)}^{3}\right)}{{\left(m \cdot \frac{1 - m}{v}\right)}^{2} + \left(1 - \frac{-m}{\frac{v}{1 - m}}\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 65.2%

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

      2. associate-*r/ 65.1%

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

      3. associate-/l* 65.1%

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

      4. associate-/r/ 65.1%

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

      5. distribute-neg-frac 65.1%

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

      6. cancel-sign-sub 65.1%

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

      7. associate-/r/ 65.1%

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

      8. associate-*r/ 65.2%

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

      9. associate-/l* 65.3%

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

   7. Simplified 65.3%

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

   8. Taylor expanded in v around 0 72.4%

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

   9. Taylor expanded in m around 0 69.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-179}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9: 88.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[m_, v_] := If[LessEqual[m, 1.0], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(m * N[((-m) / v), $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 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*l/ 88.8%

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

      6. associate-*r/ 99.7%

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

      7. *-lft-identity 99.7%

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

      8. associate-*l/ 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out 99.7%

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

      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 97.6%

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

   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/ 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* 99.8%

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

      10. *-commutative 99.8%

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

      11. distribute-rgt-out 99.8%

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

      12. associate-*r/ 99.9%

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

      13. associate-/l* 99.9%

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

      14. /-rgt-identity 99.9%

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

      15. associate-*l/ 99.9%

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

      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 v around 0 99.9%

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

      1. associate-/l* 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 0.1%

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

      1. unpow2 0.1%

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

   9. Simplified 0.1%

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

       1. add-sqr-sqrt 0.1%

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

       2. sqrt-unprod 0.1%

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

       3. sqr-neg 0.1%

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

       4. sqrt-unprod 0.0%

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

       5. add-sqr-sqrt 79.0%

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

       6. neg-mul-1 79.0%

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

       7. *-un-lft-identity 79.0%

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

       8. times-frac 79.0%

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

       9. metadata-eval 79.0%

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

   11. Applied egg-rr 79.0%

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

       1. mul-1-neg 79.0%

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

       2. associate-*r/ 79.0%

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

       3. distribute-rgt-neg-in 79.0%

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

       4. distribute-frac-neg 79.0%

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

   13. Simplified 79.0%

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

4. Final simplification 89.7%

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

Alternative 10: 27.5% 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/ 93.5%

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

   6. associate-*r/ 99.8%

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

   7. *-lft-identity 99.8%

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

   8. associate-*l/ 99.7%

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

   9. associate-*r* 99.7%

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

   10. *-commutative 99.7%

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

   11. distribute-rgt-out 99.7%

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

   12. associate-*r/ 99.8%

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

   13. associate-/l* 99.8%

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

   14. /-rgt-identity 99.8%

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

   15. associate-*l/ 99.8%

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

   16. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in m around 0 28.3%

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

   1. neg-mul-1 28.3%

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

6. Simplified 28.3%

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

7. Final simplification 28.3%

   \[\leadsto -m \]

Reproduce

herbie shell --seed 2023174 
(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))